A unified half-integral Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups
J. Pascal Gollin, Kevin Hendrey, Ken-ichi Kawarabayashi, O-joung Kwon, Sang-il Oum
aa r X i v : . [ m a t h . C O ] F e b A UNIFIED HALF-INTEGRAL ERD ˝OS-P ´OSA THEOREM FOR CYCLES INGRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS
J. PASCAL GOLLIN, KEVIN HENDREY, KEN-ICHI KAWARABAYASHI, O-JOUNG KWON,AND SANG-IL OUM
Abstract.
Erd˝os and P´osa proved in 1965 that there is a duality between the maximum size ofa packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality doesnot hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles byrelaxing packing to half-integral packing. We prove a far-reaching generalisation of the theorem ofReed; if the edges of a graph are labelled by finitely many abelian groups, then there is a dualitybetween the maximum size of a half-integral packing of cycles whose values avoid a fixed finite setfor each abelian group and the minimum size of a vertex set hitting all such cycles.A multitude of natural properties of cycles can be encoded in this setting, for example cyclesof length at least ℓ , cycles of length p modulo q , cycles intersecting a prescribed set of vertices atleast t times, and cycles contained in given Z -homology classes in a graph embedded on a fixedsurface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set offinitely many such properties. Introduction
A classical theorem of Erd˝os and P´osa [6] states that every graph contains either k vertex-disjointcycles or a vertex set of size at most O ( k log k ) that hits all cycles of G . Such a theorem does nothold if we restrict to odd cycles; Lov´asz and Schrijver (see [26]) found a class of graphs having notwo vertex-disjoint odd cycles and no small vertex set hitting all odd cycles.In the setting of odd cycles, Reed [17] obtained an analogue of the theorem of Erd˝os and P´osaby relaxing the “vertex-disjoint” condition. A half-integral packing is a set of subgraphs such thatno vertex is contained in more than two of them. Reed [17] proved that there is a function f suchthat every graph has a half-integral packing of at least k odd cycles or has a vertex set of size atmost f ( k ) hitting all odd cycles. As an easy corollary of Reed’s result, given a graph whose edgesare labelled with Z , there is a half-integral packing of at least k cycles, each of non-zero totalweight, or a vertex set of size at most f ( k ) hitting all such cycles.Very recently, Thomas and Yoo [25] extended this result to arbitrary abelian groups: they showedthat there is a function f such that given a graph whose edges are labelled by an abelian group,there is a half-integral packing of at least k cycles each of non-zero total weight, or a vertex set ofsize at most f ( k ) hitting all such cycles.Kakimura and Kawarabayashi [12] proved a different kind of strengthening of the theorem ofReed. They showed that there is a function f such that every graph contains a half-integralpacking of k odd cycles each of which intersects a prescribed set S of vertices or a vertex set of sizeat most f ( k ) hitting all such cycles. When S is the entire vertex set of the graph, this is equivalentto the theorem of Reed. This result can be encoded in the setting of graphs labelled with twoabelian groups Z and Z : in Z we label each edge incident with a vertex in S with 1, and all other Date : February 3rd, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Erd˝os-P´osa theorem, half-integral, group-labelled graph.All authors except the third author are supported by the Institute for Basic Science (IBS-R029-C1). The thirdauthor is supported by JSPS Kakenhi Grant Number JP18H05291. The fourth author is supported by the NationalResearch Foundation of Korea (NRF) grant funded by the Ministry of Education (No. NRF-2018R1D1A1B07050294). edges with 0, and in Z we label each edge with 1. The cycles which are of non-zero total weightwith respect to both of these group labellings are precisely the cycles described by Kakimura andKawarabayashi. Note that, since two groups are required for the encoding, this result is not coveredby the previously mentioned result of Thomas and Yoo.Our main theorem generalises all of these results to the setting of cycles in graphs labelled witha bounded number of abelian groups, whose values avoid a bounded number of elements of eachgroup. For an abelian group Γ and a graph G , a function γ : E ( G ) → Γ is called a Γ -labelling of G .The γ -value of a subgraph H of G is the sum of γ ( e ) over all edges e in H . For an integer m ,we write [ m ] for the set of positive integers at most m . We call a set of vertices which hits allsubgraphs in a set H a hitting set for H . Theorem 1.1.
For every pair of positive integers m and ω , there is a function f m,ω : N → N satisfying the following property. For each i ∈ [ m ] , let Γ i be an abelian group, and let Ω i be a subsetof Γ i . Let G be a graph and for each i ∈ [ m ] , let γ i be a Γ i -labelling of G , and let O be the setof all cycles of G whose γ i -value is in Γ i \ Ω i for all i ∈ [ m ] . If | Ω i | ≤ ω for all i ∈ [ m ] , then forall k ∈ N there exists either a half-integral packing of k cycles in O , or a hitting set for O of sizeat most f m,ω ( k ) . We point out that the function f m,ω does not depend on the choice of groups Γ i or the subsets Ω i .For a graph labelled with a single abelian group Γ, Wollan [29] showed that if Γ has no elementof order 2, then there are arbitrarily many vertex-disjoint cycles of non-zero γ -value, or a hittingset of bounded size for the γ -non-zero cycles. However, if Γ has an element of order 2, then as inthe case of odd cycles, this does not hold [26, 17]. Hence, even when m = ω = 1, the half-integralcondition cannot be removed in Theorem 1.1.If the number m of given groups Γ i is unbounded or the size of Ω i is unbounded in Theorem 1.1,then such a function f m,ω does not exist, and moreover, for every integer n ≥
2, a (1 /n )-integralanalogue of the Erd˝os-P´osa theorem does not hold. We discuss this in Section 3.We now present some corollaries which help illustrate the power of this theorem. In the followingcorollaries, we use the function f m,ω in Theorem 1.1. As already mentioned, the cycles in a graph G which intersect a prescribed set of vertices can be encoded as precisely the non-zero cycles withrespect to the Z -labelling which assigns value 1 to edges incident with vertices in S and 0 to allother edges. If instead each edge of G is assigned the integer that is the number of its endverticeswhich lie in S , then the cycles of total weight at least 2 t are precisely the cycles which intersectthe set S at least t times. Using multiple labellings with Z , we can encode the set of cycles whichintersect each of a bounded number of sets at least t times each. Thus we obtain our first corollary. Corollary 1.2.
Let m and t be positive integers. For each i ∈ [ m ] , let S i be a subset of the verticesof a graph G and t i ∈ [ t ] , and let O be the set of all cycles of G containing at least t i vertices of S i for all i ∈ [ m ] . Then for all k ∈ N there exists either a half-integral packing of k cycles in O , or ahitting set for O of size at most f m,t ( k ) . In fact, the construction above can be generalised, allowing us to convert a group labelling ofthe vertices of a graph to a group labelling of its edges. For an abelian group Γ and a graph G ,a Γ -vertex-labelling of G is a function γ : V ( G ) → Γ, and the γ -value of a subgraph H of G is thesum of γ ( v ) over all vertices in H . In Section 3, we will discuss in detail how such a conversionworks in general, and thus obtain the following corollary. Corollary 1.3.
Let m and ω be positive integers. For each i ∈ [ m ] , let Γ i be an abelian group andlet Ω i be a subset of Γ i . Let G be a graph, and for each i ∈ [ m ] , let γ i be a Γ i -vertex-labelling of G ,and let O be the set of all cycles of G whose γ i -value is in Γ i \ Ω i for all i ∈ [ m ] . If | Ω i | ≤ ω forall i ∈ [ m ] , then for all k ∈ N there exists either a half-integral packing of k cycles in O , or a hittingset for O of size at most f m,ω ( k ) . ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 3
Our next corollary relates to graphs labelled with a fixed finite abelian group, where we obtain asimilar result for the set of cycles of any specified value. In particular, this shows that for every pairof positive integers p and q , cycles of length p modulo q satisfy a half-integral analogue of the Erd˝os-P´osa theorem. Dejter and Neumann-Lara [4] showed that without the half-integral relaxation, theanalogous Erd˝os-P´osa type result fails for cycles of length p modulo q whenever the least commonmultiple lcm( p, q ) of p and q is divisible by 2 p (see also [29]). When this condition is not met,Gollin, Hendrey, Kwon, Oum, and Yoo [8] show that the half-integral relaxation is required for anErd˝os-P´osa type result in this setting if and only if lcm( p, q ) /p is divisible by three distinct primes. Corollary 1.4.
Let Γ be a finite abelian group and let g ∈ Γ . Let G be a graph, and γ be a Γ -labelling of G , and let O be the set of all cycles of γ -value g . Then for all k ∈ N there exists eithera half-integral packing of k cycles in O , or a hitting set for O of size at most f , | Γ |− ( k ) . Our next corollary relates to graphs embedded on a fixed compact surface, where we obtain asimilar result for the set of cycles contained in any given set of (first) Z -homology classes. Huynh,Joos, and Wollan [10] proved that for graphs embedded on a fixed surface, cycles not homologous tozero in the Z -homology group satisfy a half-integral analogue of the Erd˝os-P´osa theorem. They useda different type of graph labelling, called a directed Γ-labelling. With (undirected) Γ-labellings, wecan do the same thing with respect to Z -homology classes. Since a compact surface has a finiteabelian group as its Z -homology group, we can obtain a half-integral analogue of the Erd˝os-P´osatheorem for cycles contained in any given set of Z -homology classes. We discuss this further inSection 3. Corollary 1.5.
Let Σ be a compact surface with Z -homology group Γ and let C be a set of Z -homology classes of Σ . Let G be a graph embedded on Σ , and let O be the set of all cycles of G whose Z -homology classes are contained in C . Then for all k ∈ N there exists either a half-integralpacking of k cycles in O , or a hitting set for O of size at most f , | Γ |−|C| ( k ) . One nice feature of our main theorem is that it allows us to combine various properties of cyclestogether to obtain new results, as long as we take a bounded number of properties and can encodeeach of them with a bounded number of group labellings. Thus, we could combine any subset ofthese corollaries together and obtain a result of the same form.Huynh, Joos, and Wollan [10] obtained a result similar to our main theorem for graphs withtwo directed group labellings, where the value of an edge is inverted if it is traversed in the reversedirection. They showed that a half-integral analogue of the Erd˝os-P´osa theorem holds for cycleswhose values are non-zero in each coordinate. They conjectured that their result can be extended tographs with more than two directed labellings. Because the Γ-labellings in this paper are equivalentto directed Γ-labellings when all elements of Γ have order 2, Theorem 1.1 implies that the conjectureof Huynh, Joos, and Wollan hold for graphs with a fixed number of directed labellings with suchgroups. The conjecture otherwise remains open, although there is a large overlap between themotivation of the conjecture and the consequences of our main theorem. We discuss directed grouplabellings in more detail in Section 10.The structure of this paper is as follows. In Section 2, we introduce preliminary concepts, andwe give a high-level overview of the proof of our main theorem and present proofs of corollaries inSection 3. In Section 4, we define a packing function and provide its application. In Section 5, wedefine a concept of a clean wall which is well-behaved for each of the labellings γ i . In Section 6, weprove a key lemma to find many vertex-disjoint paths attached to a wall. In Section 7, we proveuseful lemmas on a product of abelian groups that will be used in the last step of the main theorem.Section 8 discusses how to obtain a desired cycle from a wall together with attached disjoint paths.We prove our main result in Section 9, and we discuss some open problems in Section 10. J. PASCAL GOLLIN, KEVIN HENDREY, KEN-ICHI KAWARABAYASHI, O-JOUNG KWON, AND SANG-IL OUM Preliminaries
In this paper, all graphs are undirected simple graphs having no loops and multiple edges. Forevery abelian group, we regard its operation as an additive operation and denote its zero by 0. Eventhough we work on simple graphs, all the results are extended to multigraphs; given a multigraphinstance, we can take a subdivision to produce an equivalent simple graph with group value 0 onnew edges.For an integer m , we write [ m ] for the set of positive integers at most m .Let G be a graph. We denote by V ( G ) and E ( G ) the vertex set and the edge set of G , respectively.For a vertex set A of G , we denote by G − A the graph obtained from G by deleting all the verticesin A and all edges incident with vertices in A , and denote by G [ A ] the subgraph of G inducedby A , which is G − ( V ( G ) \ A ). If A = { v } , then we write G − v for G − A . For an edge e of G ,we denote by G − e the graph obtained by deleting e . For two graphs G and H , let G ∪ H := ( V ( G ) ∪ V ( H ) , E ( G ) ∪ E ( H )) and G ∩ H := ( V ( G ) ∩ V ( H ) , E ( G ) ∩ E ( H )) . For a set G of graphs, we denote by S G the union of the graphs in G .For an integer t , a graph G is t -connected if it has more than t vertices and G − S is connectedfor all vertex sets S with | S | < t . Subdividing an edge uv in a graph G is an operation that yields a graph containing one newvertex w , and with an edge set replacing uv by two new edges, uw and wv . A graph H is a subdivision of a graph G if H can be obtained from G by subdividing edges repeatedly.Let A and B be vertex sets of G . An ( A, B ) -path is a path from a vertex in A to a vertexin B such that all internal vertices are not contained in A ∪ B . We also denote an ( A, A )-path asan A -path . For a subgraph H of G , we shortly write as an H -path for a V ( H )-path. A path is A -intersecting if it contains a vertex of A .For a graph G , we denote by V =2 ( G ) the set of all vertices of G whose degrees are not equal to 2.A corridor of a graph G is a V =2 ( G )-path of length at least 1. Remark 2.1.
Every graph in which no block is a cycle is the edge-disjoint union of its corridors.
For a family F = ( x i : i ∈ I ) we write |F | = | I | , called the size of F .2.1. Walls.
Let c, r ≥ elementary ( c, r ) -wall W c,r is the graph obtained fromthe graph on the vertex set [2 c ] × [ r ] whose edge set is { ( i, j )( i + 1 , j ) : i ∈ [2 c − , j ∈ [ r ] } ∪ { ( i, j )( i, j + 1) : i ∈ [2 c ] , j ∈ [ r − , i + j is odd } by deleting both degree 1 vertices. Figure 1.
An elementary (5 , W depicted on the left with a 3-column-slice,and depicted on the right with a (4 , V =2 ( W )-anchored.For j ∈ [ r ], the j -th row R j of W c,r is the path W c,r (cid:2)(cid:8) ( i, j ) ∈ V ( W c,r ) : i ∈ [2 c ] (cid:9)(cid:3) . For i ∈ [ c ],the i -th column C i of W c,r is the path W c,r (cid:2)(cid:8) ( i ′ , j ) ∈ V ( W c,r ) : i ′ ∈ { i − , i } , j ∈ [ r ] (cid:9)(cid:3) .A ( c, r ) -wall is a subdivision W of the elementary ( c, r )-wall. If W is a ( c, r )-wall for somesuitable integers c and r , then we say W is a wall of order min { c, r } . We call a branch vertex ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 5 corresponding to the vertex ( i, j ) of the elementary wall a nail of W , and denote by N W the set ofnails of W . Remark 2.2.
Any wall is a subdivision of a -connected planar graph. For a subgraph H of the elementary wall, we denote by H W the subgraph of W correspondingto a subdivision of H . We call R Wj or C Wi the j -th row or i -th column of W , respectively. Asubgraph W ′ of a wall W that is itself a wall is called a subwall of W . For a set S of vertices, wesay a wall W is S -anchored if N W ⊆ S . We observe the following. Remark 2.3.
Let W be a ( c, r ) -wall for integers c, r ≥ . Then W contains a ( c − , r − -subwall W ′ which is V =2 ( W ) -anchored. See Figure 1. (cid:3) For an integer c ≥
3, we call a subwall W ′ of a wall W a c -column-slice of W if the set of nailsof W ′ is exactly N W ∩ V ( W ′ ), there is a column of W ′ which is a column of W , and W ′ has exactly c columns, see Figure 1 for an example. Similarly, for an integer r ≥
3, we call a subwall W ′ of awall W an r -row-slice of W if the set of nails of W ′ is exactly N W ∩ V ( W ′ ), there is a row of W ′ which is a row of W , and W ′ has exactly r rows. Note that in an r -row-slice W ′ of W , dependingon the location, the first column of W ′ may be in the last column of W by the definition of a wall.See Figure 2 for an illustration. Figure 2.
A 3-row-slice W , a 4-row-slice W , and a 5-row-slice W of a (5 , W . Notice that the first column of W is in the first column of W but the firstcolumn of W or W is in the last column of W .Given a wall W , a W -handle is a non-trivial W -path whose endvertices are degree-2 nails of W contained in the union of the first and last column of W .Let W be a ( c, r )-wall, let W ′ be a c ′ -column-slice of W for some 3 ≤ c ′ ≤ c , and let P bea W -handle. We define the row-extension of P to W ′ in W as the unique non-trivial W ′ -pathin P ∪ S { R Wi : i ∈ [ r ] } . Note that such a P is a W ′ -handle. For a set P of vertex-disjoint W -handles, we define the row-extension of P to W ′ in W to be the set of row-extensions of the pathsin P to W ′ in W . Note that these W ′ -handles are vertex-disjoint.2.2. Linkages, separations, and tangles.
Let G be a graph. For vertex sets A and B in G , aset P of vertex-disjoint ( A, B )-paths of G is called a linkage from A to B , and its order is definedto be |P| . A separation of G is a pair ( A, B ) of subsets of V ( G ) such that G [ A ] ∪ G [ B ] = G . Its order is defined to be | A ∩ B | . We will use Menger’s theorem. Theorem 2.4 (Menger [14]) . Let A and B be vertex sets in a graph G , and k be a positive integer.Then G contains either a linkage of order k from A to B , or a separation ( A ′ , B ′ ) of order lessthan k such that A ⊆ A ′ and B ⊆ B ′ . We need a concept of a large wall dominated by a tangle, see [20, (2.3)]. For a positive integer t ,a set T of separations of order less than t is a tangle of order t in G if it satisfies the following.(1) If ( A, B ) is a separation of G of order less than t , then T contains exactly one of ( A, B )and (
B, A ).(2) If ( A , B ) , ( A , B ) , ( A , B ) ∈ T , then G [ A ] ∪ G [ A ] ∪ G [ A ] = G . J. PASCAL GOLLIN, KEVIN HENDREY, KEN-ICHI KAWARABAYASHI, O-JOUNG KWON, AND SANG-IL OUM
Let W be a wall of order g with g ≥ G . Let T W be the set of all separations ( A, B )of G of order less than g such that G [ B ] contains a row of W . By the following simple lemma, wemay replace the row with the column. The proof in [19] is for the grid but one can easily modifyit for the wall. Lemma 2.5 (Robertson and Seymour [19, (7.1)]) . Let W be a wall of order g in a graph G .Let ( A, B ) be a separation of order less than g . Then G [ B ] contains a row of W if and only if itcontains a column of W . Kleitman and Saks (see [19, (7.3)]) showed that T W is a tangle of order g . A tangle T in G dominates the wall W if T W ⊆ T . Theorem 2.6 (Robertson, Seymour, and Thomas [20]) . There exists a function f . : N → N suchthat if g ≥ is an integer and T is a tangle in a graph G of order at least f . ( g ) , then T dominatesa ( g, g ) -wall W in G . We will show that if a tangle dominates a wall W , then it also dominates every large subwallof W . We first prove the following lemma. Lemma 2.7.
Let W be a wall in a graph G and let S be a subset of V ( G ) of size exactly t . For eachcolumn C Wx and row R Wy of W , there are no more than t nails of W which belong to componentsof W − S that do not intersect C Wx ∪ R Wy .Proof. We proceed by induction on t . The statement is trivial if t = 0. We may assume that G = W .Let S =: { s i : i ∈ [ t ] } and let T := V ( C Wx ∪ R Wy ). Suppose there is a vertex v in S \ N W , and let P be the N W -path in W containing v . If both or neither of the endvertices of P are in componentsof W − S that intersect T , then we may apply the inductive hypothesis to S \ { v } . Otherwise,replacing v in S with the unique endvertex of P which is in a component of W − S that intersects T does not decrease the number of nails in components of W − S that do not intersect T . Hence, wemay assume that S ⊆ N W .For each i ∈ [ t ], let c ( i ) be the integer such that s i is in C Wc ( i ) and let r ( i ) be the integer suchthat s i is in R Wr ( i ) . For each i ∈ [ t ], let S ci be the set of nails v of W in C Wc ( i ) − S such that the( v, R Wy )-subpath of C Wc ( i ) contains the vertex s i , and let S ri be the set of nails v of W in R Wr ( i ) − S such that the ( v, C Wx )-subpath of R Wr ( i ) contains the vertex s i . Note that for every nail v of W , if v is in a component of W − S not intersecting T , then there exist i, j ∈ [ t ] such that v ∈ S ci ∩ S rj .Also note that | S ci ∩ S rj | ≤
2, and that if | S ci ∩ S rj | > i = j , then | S cj ∩ S ri | = 0. Furthermore,for i ∈ [ t ], the nails in S ci ∩ S ri are in ( C Wc ( i ) ∩ R Wr ( i ) ) − s i , so | S ci ∩ S ri | ≤
1. It follows that thenumber of nails of W which are in components of W − S that do not intersect C Wx ∪ R Wy is atmost 2 (cid:0) t (cid:1) + t = t . (cid:3) Lemma 2.8.
Let w ≥ t ≥ be integers, let W be a wall of order w in a graph G , and let T be atangle dominating W . If W ′ is a subwall of W of order t and | N W ′ ∩ N W | > (2 t − t − , then T dominates W ′ .In particular, if W ′ is N W -anchored, then T dominates W ′ .Proof. Suppose for a contradiction that T does not dominate W ′ . Then G has a separation ( A, B )of order less than t such that G [ B ] contains some row R ′ of W ′ and ( A, B ) / ∈ T . The order of T isat least w , so T contains ( B, A ) and therefore G [ A ] contains some row R of W . Let S := A ∩ B .Since W has more than | S | columns and R intersects each of them, G [ A ] contains some column C of W , and similarly G [ B ] contains some column C ′ of W ′ . By Lemma 2.7 applied to W , thereare at most ( t − nails of W in components of G − S which do not intersect R ∪ C . Similarly,there are at most ( t − nails of W ′ in components of G − S which do not intersect R ′ ∪ C ′ . ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 7
Since | N W ′ ∩ N W | > (2 t − t − ≥ t − + | S | , there is a vertex in ( N W ′ ∩ N W ) \ S thatis in a component of G − S intersecting R ∪ C and also in a component of G − S intersect-ing R ′ ∪ C ′ . However, every component of G − S intersecting R ∪ C is in G [ A \ B ] and everycomponent of G − S intersecting R ′ ∪ C ′ is in G [ B \ A ], a contradiction. (cid:3) Groups.
Let Γ i be a group for each i ∈ [ m ]. We refer to the direct product of these groupsby Q i ∈ [ m ] Γ i and denote by π j the projection map from Q i ∈ [ m ] Γ i to Γ j for j ∈ [ m ]. For an element g of Q i ∈ [ m ] Γ i , we refer to the image π i ( g ) as the i -th coordinate of g . When we say Γ = Q i ∈ [ m ] Γ i isa product of groups, we implicitly use this notation.For a non-empty set of elements S = { a i : i ∈ [ t ] } in a group Γ, we denote by h S i or h a i : i ∈ [ t ] i the subgroup generated by S , which is the intersection of all subgroups of Γ containing S .2.4. Group-labelled graphs.
Let Γ be an abelian group. A Γ -labelled graph is a pair of a graph G and a function γ : E ( G ) → Γ. We say that γ is a Γ -labelling of G . A subgraph of a Γ-labelledgraph ( G, γ ) is a Γ-labelled graph (
H, γ ′ ) such that H is a subgraph of G and γ ′ is the restrictionof γ to E ( H ). By a slight abuse of notation, we may refer to this Γ-labelled graph by ( H, γ ).For a Γ-labelled graph (
G, γ ) and a subgraph H ⊆ G , we define γ ( H ) as P e ∈ E ( H ) γ ( e ), whichwe call the γ -value of H . Note that this definition implies that the γ -value of the empty subgraphis 0. We say that a subgraph H is γ -non-zero if γ ( H ) = 0, and otherwise, we call it γ -zero .We will often consider the special case where Γ is the product Q i ∈ [ m ] Γ i of m abelian groups fora positive integer m . In this case, we denote by γ i the composition of γ with the projection to Γ i .A Γ-labelled graph ( G, γ ) is γ -bipartite if every cycle of G is γ -zero.We frequently take a subgroup Λ of Γ and consider a new labelling using the quotient group Γ / Λ.For a Γ-labelled graph (
G, γ ) and a subgroup Λ of Γ, the Γ / Λ-labelling λ defined by λ ( e ) := γ ( e ) + Λfor all edges e ∈ E ( G ) is the induced (Γ / Λ) -labelling of ( G, γ ).We will use the following duality theorem between packing and covering of γ -non-zero A -paths. Theorem 2.9 (Wollan [28]) . Let k be a positive integer, let Γ be an abelian group, let ( G, γ ) be a Γ -labelled graph, and let A ⊆ V ( G ) . Then G contains k vertex-disjoint γ -non-zero A -paths or thereexists a set X ⊆ V ( G ) of size at most f . ( k ) := 50 k such that G − X has no γ -non-zero A -paths. Let x be a vertex of G and let δ ∈ Γ be an element of order 2. For each edge e of G , let γ ′ ( e ) = ( γ ( e ) + δ if e is incident with x , γ ( e ) otherwise.We say that γ ′ is obtained from γ by shifting by δ at x . Observe that this shift does not changethe weight sum of a cycle because δ + δ = 0. We say two Γ-labellings γ and γ of G are shifting-equivalent if γ can be obtained from γ by a sequence of shifting operations.The following lemma asserts that for γ -bipartite graphs we can find a shifting-equivalent Γ-labelling γ ′ in which every corridor is γ ′ -zero. Similar ideas appear in Geelen and Gerards [7]. Lemma 2.10.
Let Γ be an abelian group, let ( G, γ ) be a Γ -labelled graph and let H ⊆ G be asubdivision of a -connected graph ˆ H . If H is γ -bipartite, then γ is shifting-equivalent to a Γ -labelling γ ′ such that every corridor of H is γ ′ -zero.Proof. Let T be a spanning tree of ˆ H rooted at some r ∈ V ( ˆ H ). It is enough to find a Γ-labelling γ ′ of G which is shifting-equivalent to γ , such that all corridors of H corresponding to edges in T are γ ′ -zero, because H is γ ′ -bipartite. Choose a Γ-labelling γ ′ shifting-equivalent to γ and a subtree T ′ of T containing r such that all corridors of H corresponding to edges in T ′ are γ ′ -zero, and subjectto these conditions, | V ( T ′ ) | is maximised.Suppose that T ′ = T . Then there is an edge vw of T such that v ∈ V ( T ′ ) and w / ∈ V ( T ′ ). Let P be the corridor of H corresponding to the edge vw . Since ˆ H is 3-connected, there is a cycle O J. PASCAL GOLLIN, KEVIN HENDREY, KEN-ICHI KAWARABAYASHI, O-JOUNG KWON, AND SANG-IL OUM in H − E ( P ) containing v and w . Let O and O denote the distinct cycles in O ∪ P containing P .Since H is γ ′ -bipartite, we have that γ ′ ( P ) + γ ′ ( P ) = γ ′ ( O ) + γ ′ ( O ) − γ ′ ( O ) = 0, and hence γ ′ ( P )is an element of order at most 2. Let γ ′′ be a Γ-labelling of G obtained from γ ′ by shifting by γ ′ ( P )at w . Let T ′′ = T [ V ( T ′ ) ∪ { w } ]. Then all corridors of H corresponding to edges of T ′′ are γ ′′ -zero,contradicting the choice of γ ′ and T ′ . (cid:3) Discussion
Proof Sketch.
We now sketch the proof of Theorem 1.1, which will proceed by induction on k .We consider the group Γ := Q i ∈ [ m ] Γ i and a single Γ-labelling, which simplifies the arguments wepresent and in particular allows us to consider quotient groups. The goal will be to show that ifthe smallest hitting set T for the cycles in O is sufficiently larger than f m ( k − k cycles in O . To construct this packing, in Section 4, we first find atangle whose order is correlated with | T | /f m ( k − G . In particular, this wall will have the property that no cycle in O can be separatedfrom the nails of the wall by deleting a small set of vertices. Our strategy will be to find disjointsets of γ -non-zero paths and use the structure of the wall to connect them up to form cycles. Butbefore we begin to do this, in Section 5 we find a subwall W of the original wall such that the N W -paths in W have some nice homogeneity properties with respect to the group labelling. It wouldbe simplest if we could guarantee that all N W -paths in W were γ -zero, but this is not feasible witharbitrary abelian groups. Instead, we deal separately with the factors Γ i of Γ for which we canguarantee that all N W -paths in W are γ i -zero, and the factors of Γ for which we cannot find anylarge subwall of W with this property.Applying Theorem 2.9, we can find for every factor Γ i of Γ a large set of disjoint N W -paths whichare γ i -non-zero. The difficulty here lies in combining these paths together such that for all i ∈ [ m ]the total γ i -value is not in Ω i . To achieve this, in Section 6 we show how to iteratively find sets ofdisjoint paths which are non-zero with respect to a quotient group defined in terms of the previouslyconstructed paths, and link them up to the boundary of a subwall of W . In Section 7, we analyseconditions under which we can find a set of elements from a product of abelian groups whose sumavoids a finite set Ω i in each coordinate. In Section 8, we discuss how to combine the disjoint setsof paths into cycles using the wall. This allows us to find a half-integral packing of k cycles whose γ i -values are in Γ i \ Ω i for every factor Γ i of Γ for which the N W -paths in W are γ i -zero.To deal with each remaining factor Γ i , we observe that since no large subwall of W is γ i -bipartite,every large subwall of W contains a γ i -non-zero cycle. In fact, we iteratively find disjoint cycles in W which are non-zero with respect to quotient groups defined in terms of the previously constructedcycles. We link these cycles up to the half-integral packing of k -cycles we have constructed, and byrerouting through them, transform each cycle in our half-integral packing into a cycle in O .3.2. Obstructions for integral and half-integral Erd˝os-P´osa type results.
One funda-mental obstruction to Erd˝os-P´osa type results, known as the Escher wall, is due to Lov´asz andSchrijver (see [26]). An
Escher wall of height n is obtained from an ( n, n )-wall W by adding afamily ( P i : i ∈ [ n ]) of vertex-disjoint W -paths, such that for each i ∈ [ n ], one endvertex of P i isin R W ∩ C Wi and the other is in R Wn ∩ C Wn +1 − i . See Figure 3 for an illustration. They observedthat there are no two disjoint cycles in such an Escher wall which each contain an odd number ofpaths in ( P i : i ∈ [ n ]), but for any set S of at most n − P i : i ∈ [ n ]). Using this construction, Thomassen [26] arguedthat an analogue of the Erd˝os-P´osa theorem does not hold for odd cycles. It can be further used toshow that the same holds for cycles of length ℓ modulo m whenever m is an even positive integerand ℓ is an odd integer with 0 < ℓ < m , see Wollan [29].Many previous half-integral Erd˝os-P´osa type results have relied on characterising Escher walls asthe fundamental obstructions to finding integral packings of certain classes of cycles. Half-integral ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 9
Figure 3.
An Escher wall of height 7.packings are then obtained using the structure of the Escher Wall. In our setting, the Escher Wallis not the only possible obstruction which can arise. In fact, unlike Escher walls, the followingtype of obstruction can occur even in planar graphs. For this reason, some of the structural resultswhich have been used to obtain other half-integral Erd˝os-P´osa type results are unlikely to be usefulin our setting. For example, we do not use the flat wall theorem of Robertson and Seymour [22, 21]in this paper, and there is no obvious way to significantly simplify our proofs by doing so.
Proposition 3.1.
Let G be the n × n -grid, and let O be the set of cycles of G which contain atleast one edge of the top row R t of G , at least one edge of the bottom row R b of G , and at least oneedge from the leftmost column C ℓ of G , and let O ′ be the set of cycles of G which contain exactlyone edge of the top row R t of G , exactly one edge of the bottom row R b of G , and exactly one edgefrom the leftmost column C ℓ of G . Then every pair of cycles in O intersect, but there is no hittingset for O ′ of size less than ( n − / .Proof. We may assume that n ≥
3. Note that the graph G ′ obtained from G by adding a ver-tex z adjacent to every vertex of R t ∪ R b ∪ C ℓ is planar. Suppose for contradiction that there aredisjoint cycles O and O in O , and let H be the component of G − O containing O . Since C ℓ is connected O intersects C ℓ − ( R t ∪ R b ), there is a vertex v in ( O ∩ C ℓ ) − ( R t ∪ R b ) adjacentto a vertex of H , and similarly a vertex v in O ∩ R t adjacent to a vertex of H and a vertex v in O ∩ R t adjacent to a vertex of H . Contracting C to a triangle on { v , v , v } and H to a singlevertex, we find a K -minor in G ′ , contradicting Wagner’s Theorem.Now consider S ⊆ V ( G ) of size less than ( n − /
2. Note that there are two adjacent columns C i and C i +1 which do not intersect S , and likewise two adjacent rows R j and R j +1 which do notintersect S . It is easy to see that the subgraph of G induced on the vertices in C i ∪ C i +1 ∪ R j ∪ R j +1 contains a cycle in O ′ . (cid:3) As an example, consider an n × n -grid in which all edges on the top row are subdivided exactly 14times, all edges of the bottom row are subdivided exactly 69 times, all edges of the leftmostcolumn are subdivided exactly 20 times, and all other edges are subdivided exactly 104 times. ByProposition 3.1, there are no two vertex-disjoint cycles of length 1 mod 105, and no hitting set forthese cycles of size less than ( n − / n × n -grid,let S be the set of all vertices on the top row, let S be the set of all vertices of the bottom row,and let S be the set of all vertices of the leftmost column. By Proposition 3.1, there are no twovertex-disjoint cycles each containing at least one vertex from each of S , S , S , and no hitting setfor these cycles of size less than ( n − / Proposition 3.2.
Let t and c be positive integers, let Γ be an abelian group and let Ω be a subsetof Γ such that there is an element g ∈ Γ and an integer d > ( c − t such that d is the minimuminteger greater than for which dg / ∈ Ω . Then there is a graph G with Γ -labelling γ such that, forthe set O of all cycles of G whose γ -values are not in Ω , every c cycles in O share a commonvertex, but there is no hitting set for O of size less than t .Proof. The c = 1 case is trivial, so we may assume c ≥
2. Let n := ⌈ cd/ ( c − ⌉ −
1, let G := K n ,and let γ be the Γ-labelling assigning g to every edge of G . By construction, every cycle in O haslength greater than ( c − n/c , and so every c cycles in O share a common vertex. However, everycycle of length d in G is in O , so the smallest hitting set for O has size n − ( d − > t . (cid:3) As an example, consider an infinite abelian group which contains arbitrarily large finite cyclicsubgroups. One consequence of Proposition 3.2 is that there is no half-integral Erd˝os-P´osa typeresult for cycles of weight zero in graphs labelled with such a group.As another consequence, we obtain a lower bound on the functions mentioned in Theorem 1.1which depends on both m and ω . Corollary 3.3.
For any function f m,ω as in Theorem 1.1, we have f m,ω (3) > (2 + mω ) / .Proof. Consider for each i ∈ [ m ] the group Γ i := Z and the subset Ω i := [2 + ωi ] \ [2 + ω ( i − c = 3, g = (1 : i ∈ [ m ]), and Ω = S i ∈ [ m ] { g ′ ∈ Γ : π i ( g ′ ) ∈ Ω i } from Proposi-tion 3.2. (cid:3) Relating vertex-labellings to edge-labellings.
We now demonstrate how to convert agroup labelling of the vertices of a graph to a group labelling of its edges. Corollary 1.3 followseasily from Theorem 1.1 after applying the following lemma to each of the vertex-labellings itmentions.
Lemma 3.4.
Let Γ be an abelian group, let Ω ⊆ Γ be a finite subset, let G be a graph, and let γ be a Γ -vertex-labelling of G . Then there is a group Γ ′ , a subset Ω ′ ⊆ Γ ′ with | Ω ′ | = | Ω | , and a Γ ′ -labelling γ ′ of G such that for every cycle O of G , we have γ ( O ) ∈ Ω if and only if γ ′ ( O ) ∈ Ω ′ .Proof. Let Γ ′′ = h Ω ∪ { γ ( v ) : v ∈ V ( G ) }i . By the fundamental theorem of finitely generated abeliangroups, there exist an integer m and an isomorphism ϕ from Γ ′′ to a product Q i ∈ [ m ] Γ i , whereeach Γ i is either Z , or a cyclic group Z n i of order n i . For i ∈ [ m ], let Γ ′ i := Z n i if the order of Γ i isfinite, and Γ ′ i := Γ i otherwise. Let Γ ′ := Q i ∈ [ m ] Γ ′ i . For i ∈ [ m ], let e i denote the element of Γ ′′ suchthat π j ( ϕ ( e i )) = 1 if i = j , and π j ( ϕ ( e i )) = 0 if i = j . Then Γ ′′ = h e i : i ∈ [ r ′ ] i . For i ∈ [ m ], let e ′ i denote the element of Γ ′ such that π j ( e ′ i ) = 1 if i = j , and π j ( e ′ i ) = 0 if i = j . For each j ∈ [2], wedefine a homomorphism ψ j from Γ ′′ to Γ ′ by setting ψ j ( e i ) := je ′ i on the generators. Note that ψ is injective since the kernel of ψ is trivial, that the image of ψ is 2Γ ′ , and that ψ (2 g ) = ψ ( g )for all g ∈ Γ ′′ . We define Ω ′ := ψ (Ω) and a Γ ′ -labelling γ ′ of G for an edge e = vw of G bysetting γ ′ ( e ) = ψ (cid:0) γ ( v ) + γ ( w ) (cid:1) . Note that for a cycle O of G , we have that γ ′ ( O ) = X e ∈ E ( O ) γ ′ ( e ) = X vw ∈ E ( O ) ψ ( γ ( v ) + γ ( w )) = ψ X v ∈ V ( O ) γ ( v ) = ψ ( γ ( O )) . Hence, the result follows from the injectivity of ψ . (cid:3) Graphs embedded on a surface.
We now discuss how our result applies to graphs em-bedded on a surface, where we consider the first homology group with coefficients in Z . Huynh,Joos, and Wollan [10, Proposition 5] demonstrated that given a graph G embedded on a surfacewhose Z -homology group is Γ, there is a directed Γ-labelling of G so that the set of cycles in G that are homologous to zero is exactly the set of cycles having group value 0 in the labelling. This ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 11 allowed them to obtain for graphs embedded on a surface a half-integral Erd˝os-P´osa result for thenon-null-homologous cycles of the embedding. Our result works in essentially the same way.A graph H is called even if every vertex of H has even degree. For a graph G , let C ( G ) denote the cycle space of G over Z , that is the vector space of all even subgraphs H of G with the symmetricdifference as the operation. Proposition 3.5.
Let G be a graph, let Γ be an abelian group, and let φ : C ( G ) → Γ be a group ho-momorphism. Then there is a Γ -labelling γ of G such that γ ( H ) = φ ( H ) for every even subgraph H of G .Proof. Without loss of generality, we may assume that G is connected. Let T be a spanningtree of G . For each edge e ∈ E ( G ) \ E ( T ), let C e,T denote the unique cycle in T + e . We de-fine γ ( e ) := 0 for each e ∈ E ( T ) and γ ( e ) := φ ( C e,T ) for each e ∈ E ( G ) \ E ( T ). The statement nowtrivially follows, because the set { C e,T : e ∈ E ( G ) \ E ( T ) } forms a basis of the cycle space (see [5,Theorem 1.9.5]). (cid:3) Now for a graph G embedded in a surface Σ, the map assigning each even subgraph its Z -homology class is a group homomorphism from C ( G ) to the Z -homology group of Σ. Hence,Corollary 1.5 follows with Proposition 3.5 from Theorem 1.1.Note that for a closed orientable surface, the set of simple closed curves homologous to zero forthe Z -homology is exactly the same as for the Z -homology. This follows the universal coefficienttheorem (see [9]), which allows us to relate the Z -homology with the Z -homology by taking allcoefficients modulo 2. We then apply a classical result which states that no simple closed curve has Z -homology class kh for any integer k ≥ h of the Z -homology (see forexample [23]). Hence, in the case of graphs embedded on closed orientable surfaces, we recover theresult of Huynh, Joos and Wollan for non-null-homologous cycles.4. Packing functions and hitting sets
In this section, we introduce the concept of packing functions as a tool to generalise the ideas ofboth integral and half-integral packings of subgraphs, which enables us to discuss these and similarideas in a unified way.For a function ν from the set of subgraphs of a graph G to the set of non-negative integers, wesay that • ν is monotone if ν ( H ) ≤ ν ( H ′ ) whenever H is a subgraph of H ′ , • ν is additive if ν ( H ∪ H ′ ) = ν ( H ) + ν ( H ′ ) whenever H and H ′ are vertex-disjoint subgraphsof G , and • ν is a packing function for G if it is monotone and additive.Now let ν be a packing function for a graph G . For a subgraph H ⊆ G , we say a set T ⊆ V ( H )is an ν -hitting set for H if ν ( H − T ) = 0. We define τ ν ( H ) as the size of a smallest ν -hitting setof H . Note that in the traditional sense of the word, a ν -hitting set of G is a hitting set for theminimal subgraphs H ⊆ G for which ν ( H ) ≥ H of G to the maximum number of vertex-disjointcycles in H is a packing function of G .The following lemma argues that if ν ( G ) is small but G has no small ν -hitting set, then everyminimum ν -hitting set induces a tangle of large order. Similar arguments for specific packingfunctions appear many times in the literature, see [10] and [18] for instance. Lemma 4.1.
Let ν be a packing function for a graph G and let T ⊆ V ( G ) be a minimum ν -hittingset for G of size t . Let T T be the set of all separations ( A, B ) of G of order less than t/ suchthat | B ∩ T | > t/ . If τ ν ( H ) ≤ t/ whenever H is a subgraph of G with ν ( H ) < ν ( G ) , then T T is a tangle of order ⌈ t/ ⌉ . Proof.
First, we show the following claim.
Claim.
Let
X, Y ⊆ T be disjoint sets with | X | = | Y | ≥ t/ . Then there is a linkage in G from X to Y of order | X | containing no vertex in Z := T \ ( X ∪ Y ) .Proof. Suppose for a contradiction that there is no such linkage. By Menger’s theorem appliedto G − Z , there is a separation ( A, B ) of G of order strictly less than | X | + | Z | with Z ⊆ A ∩ B , X ⊆ A , and Y ⊆ B . Let S := A ∩ B . Now observe that ν ( G − ( A ∪ T )) + ν ( G − ( B ∪ T )) = ν ( G − ( S ∪ T )) ≤ ν ( G − T ) = 0 , and so ν ( G − ( A ∪ T )) = ν ( G − ( B ∪ T )) = 0. Hence ν ( G − B ) = ν ( G − B ) + ν ( G − ( A ∪ T )) = ν ( G − B ) + ν ( G − ( A ∪ Y )) = ν ( G − ( S ∪ Y )) , and so by the minimality of T and the fact that | S ∪ Y | ≤ | S | + | Y | < | X | + | Z | + | Y | = | T | wehave that ν ( G − B ) ≥
1. By symmetry ν ( G − A ) ≥
1, and since ν ( G − A ) + ν ( G − B ) ≤ ν ( G ),both ν ( G − A ) and ν ( G − B ) are strictly less than ν ( G ).By the assumption, τ ν ( G − A ) , τ ν ( G − B ) ≤ t/
12. Let T A and T B be ν -hitting sets of minimumsize for G − A and G − B , respectively. Then ν ( G − ( T A ∪ T B ∪ S )) = ν ( G − ( T A ∪ A )) + ν ( G − ( T A ∪ B )) = 0 , but | T A | + | T B | + | S | ≤ ( t/
6) + | S | ≤ | Y | + | S | < | T | , contradicting the assumption that T is a min-imum ν -hitting set. (cid:3) Let (
A, B ) be a separation of order less than t/ | B ∩ T | ≥ | A ∩ T | , and let S := A ∩ B .Clearly, ( B, A ) / ∈ T T . Suppose for a contradiction that ( A, B ) / ∈ T T and hence A \ B contains atleast t/ T . Since B \ A contains at least as many vertices of T as A \ B does, by theclaim there is a linkage of size ⌈ t/ ⌉ in G from A \ B to B \ A , contradicting the assumption onthe order of ( A, B ). Hence, (
A, B ) ∈ T T .Note that | T ∩ A | < t/ A, B ) ∈ T T . Hence for ( A , B ) , ( A , B ) , ( A , B ) ∈ T T we havethat | T ∩ ( A ∪ A ∪ A ) | < t , and hence G [ A ] ∪ G [ A ] ∪ G [ A ] = G . Thus we conclude that T T isa tangle of order ⌈ t/ ⌉ . (cid:3) Let us now turn our attention to packing functions ν for a Γ-labelled graph ( G, γ ) for an abeliangroup Γ. The following lemma is useful for converting between γ -non-zero cycles and γ -non-zeropaths. We will appeal to it in the final lemma of this section, and again in Lemma 6.3. Lemma 4.2.
Let Γ be an abelian group, let ( G, γ ) be a Γ -labelled graph, let O be a γ -non-zerocycle in G , and let T ⊆ V ( G ) . If G contains three vertex-disjoint ( V ( O ) , T ) -paths P , P , P ,then H := O ∪ P ∪ P ∪ P contains a γ -non-zero T -path.Proof. We may assume that | V ( P i ) ∩ V ( O ) | = | V ( P i ) ∩ T | = 1 for all i ∈ [3] by taking a subpath ifnecessary. For i ∈ [3], let Q i and Q ′ i be the two paths in H each having S j ∈ [3] \{ i } V ( P j ) ∩ T as itsset of endvertices, where Q ′ i is the path that is disjoint from P i . Now, X i ∈ [3] (cid:0) γ ( Q i ) − γ ( Q ′ i ) (cid:1) = 2 · X i ∈ [3] ( γ ( P i ) − γ ( P i )) + 2 · γ ( O ) − γ ( O ) = γ ( O ) = 0 . Hence, for some i ∈ [3], one of the paths Q i or Q ′ i is γ -non-zero. And since this path is the edge-disjoint union of T -paths, it contains a γ -non-zero T -path, as desired. (cid:3) Given an abelian group Γ and a Γ-labelled graph (
G, γ ), we are interested in the packing func-tion ν for G which maps a subgraph H of G to the size of the largest half-integral packing of thetype of cycles of H we are considering. To prove our main result, we will need the following toolfor finding disjoint sets of paths in G which are non-zero with respect to the induced labelling ofcertain quotient groups, which we will later construct. ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 13
Lemma 4.3.
Let u , k be positive integers such that f . ( k ) < u − . Let Γ be an abelian group,let ( G, γ ) be a Γ -labelled graph, and let ν be a packing function for G such that • every minimal subgraph H of G with ν ( H ) ≥ is a γ -non-zero cycle, • τ ν ( H ) ≤ u for every subgraph H of G with ν ( H ) < ν ( G ) , and • τ ν ( G ) ≥ u .Let T ⊆ V ( G ) be a minimum ν -hitting set for G and let N ⊆ V ( G ) such that for every S ⊆ V ( G ) of size less than u , there is a component of G − S containing a vertex of N and at least u verticesof T . Then G contains k vertex-disjoint γ -non-zero N -paths.Proof. Suppose that G does not contain k vertex-disjoint γ -non-zero N -paths. By Theorem 2.9,there exists S ⊆ V ( G ) of size less than u − γ -non-zero N -paths. Since | S | < u ≤ τ ν ( G ),we have that ν ( G − S ) ≥
1, so G − S has a γ -non-zero cycle O with ν ( O ) ≥
1. By Lemma 4.2, G − S does not have three vertex-disjoint ( V ( O ) , N )-paths. By Menger’s theorem applied to G − S , thereexists S ′ ⊆ V ( G ) of size at most | S | + 2 separating O from N . Since | S ′ | < u , by the given assump-tion on N , the graph G − S ′ has a component H containing a vertex of N and at least 4 u verticesof T . Now ν ( H ) ≤ ν ( H ∪ O ) − ν ( O ) < ν ( H ∪ O ) ≤ ν ( G ), so there is a ν -hitting set T H for H of sizeat most 3 u . Let T ′ := T H ∪ S ′ ∪ ( T \ V ( H )), and observe that | T ′ | ≤ | T H | + | S ′ | + | T | − u < | T | .But ν ( G − T ′ ) ≤ ν ( G − ( S ′ ∪ V ( H ) ∪ T )) + ν ( H − T H ) = 0, contradicting the assumption that T is a minimum ν -hitting set. (cid:3) Clean walls
In the proof of our main theorem in Section 9, we will apply Theorem 2.6 and Lemma 4.1 toconstruct a wall W in a group-labelled graph. However, it will be useful to move to a large subwallof W which has some nice homogeneity properties. For this purpose, we introduce the followingnotion of cleanness.Let Γ = Q i ∈ [ m ] Γ i be a product of m abelian groups and let ( G, γ ) be a Γ-labelled graph. Givena subset Z ⊆ [ m ] and an integer ℓ , we say that a wall W in G is ( γ, Z, ℓ ) -clean if(1) every N W -path in W is γ i -zero for all i ∈ Z , and(2) W has no ( ℓ, ℓ )-subwall which is γ i -bipartite for all i ∈ [ m ] \ Z . Lemma 5.1.
Let
Γ = Q i ∈ [ m ] Γ i be a product of m abelian groups, let ( G, γ ) be a Γ -labelled graph,let ψ : { } ∪ [ m + 1] → N ≥ be a function, and let W be a wall of order ψ (0) + 2 in G . Then thereexist a Γ -labelling γ ′ of G shifting-equivalent to γ , a subset Z of [ m ] , and a ( γ ′ , Z, ψ ( | Z | + 1) + 2) -clean V =2 ( W ) -anchored ( ψ ( | Z | ) , ψ ( | Z | )) -subwall of W .Proof. Let Z be a maximal subset of [ m ] such that there is a ( ψ ( | Z | ) + 2 , ψ ( | Z | ) + 2)-subwall W ′ of W which is γ i -bipartite for all i ∈ Z . Such a set Z exists because Z := ∅ satisfies the requirement.Since Z is maximal, there is no j ∈ [ m ] \ Z such that W ′ has a ( ψ ( | Z | + 1) + 2 , ψ ( | Z | + 1) + 2)-subwall which is γ j -bipartite.Among all Γ-labellings γ ′ of G shifting-equivalent to γ , we choose γ ′ maximising the number ofelements i ∈ Z such that all corridors of W ′ are γ ′ i -zero. If there is i ∈ Z such that some corridorof W ′ is not γ ′ i -zero, then Lemma 2.10 applied to Γ i yields the Γ-labelling γ ′′ for which everycorridor of W ′ is γ ′′ i -zero, thus contradicting the choice of γ ′ . Thus all corridors of W ′ are γ ′ i -zerofor all i ∈ Z .By Remark 2.3, W ′ has a V =2 ( W ′ )-anchored ( ψ ( | Z | ) , ψ ( | Z | ))-subwall W ′′ . Then W ′′ is V =2 ( W )-anchored since V =2 ( W ′ ) ⊆ V =2 ( W ). Now the property (1) holds since every N W ′′ -path in W ′′ is acorridor of W ′ . (cid:3) In a sense, the notion of cleanness helps us to generalise the ideas of Thomassen [26] who provedthe following result.
Proposition 5.2 (Thomassen [26]) . There exists a function w . : N → N satisfying the following.Let t and w ≥ be integers, let Γ be an abelian group generated by an element of order at most t ,and let ( W, γ ) be a Γ -labelled wall of order w . ( t, w ) . Then W contains a ( w, w ) -subwall W ′ suchthat γ ( P ) = 0 for all corridors P of W ′ . We extend Proposition 5.2 to a group generated by a fixed number of generators.
Lemma 5.3.
There exists a function w . : N → N satisfying the following. Let q , t and w ≥ beintegers, let Γ be an abelian group generated by q elements each of order at most t , and let ( W, γ ) bea Γ -labelled wall of order w . ( q, t, w ) . Then W contains a ( w, w ) -subwall W ′ such that γ ( P ) = 0 for all corridors P of W ′ .Proof. We define • w . (1 , t, w ) := w . ( t, w ), and • w . ( q, t, w ) := w . ( t, f . ( q − , t, w )) for all integers q ≥ q with Proposition 5.2 as the base case. So let q ≥
2, andlet Γ = h{ x i : i ∈ [ q ] }i for a suitable set of q generators each of order at most t . Let Γ := h x q i and Γ := h{ x i : i ∈ [ q − }i , and let γ ′ be a Γ -labelling of W such that for every edge e of W , wehave γ ( e ) + γ ′ ( e ) ∈ Γ . By Proposition 5.2, W has a ( w . ( q − , t, w ) , w . ( q − , t, w ))-subwall W ′ such that γ ( P ) ∈ Γ for all corridors P of W ′ . By the induction hypothesis, W ′ has a ( w, w )-subwall W ′′ such that γ ( P ) = 0 for all corridors P of W ′′ . Note that as W ′ is a subwall of W , allcorridors of W ′′ in W ′ are corridors of W ′′ in W . (cid:3) The following variation allows us to take advantage of our notion of cleanness and will be neededfor Lemma 8.1.
Corollary 5.4.
There exists a function w . : N → N satisfying the following. Let q , t and w ≥ be integers, let Γ be an abelian group, and let Λ be a subgroup of Γ generated by q elements eachof order at most t , and let ( W, γ ) be a Γ -labelled wall of order w . ( q, t, w ) . If γ ( O ) ∈ Λ for allcycles O of W , then W contains a γ -bipartite ( w, w ) -subwall.Proof. Let w . ( q, t, w ) := w . ( t q , t q , w ). For each g in Λ ∩ σ ( g ) be an element of Γ suchthat 2 σ ( g ) = g . Let ˆΛ be the subgroup of Γ generated by S := (Λ \ ∪ { σ ( g ) : g ∈ Λ ∩ } . Notethat Λ ⊆ ˆΛ, that | S | ≤ | Λ | ≤ t q , and that each element in S has order at most 2 | Λ | ≤ t q . We willshow that there is a Γ-labelling γ ′ shifting-equivalent to γ such that γ ′ ( P ) ∈ ˆΛ for every corridor P of W .The wall W is a subdivision of some 3-connected planar graph ˆ H . Let T be a spanning tree of ˆ H ,rooted at an arbitrary vertex r . Choose a Γ-labelling γ ′ shifting-equivalent to γ and a subtree T ′ of T containing r such that γ ′ ( P ) ∈ ˆΛ for all corridors P of W corresponding to edges in T ′ , andsubject to these conditions, | V ( T ′ ) | is maximised.Suppose that T ′ = T . Then there is an edge vw of T such that v ∈ V ( T ′ ) and w / ∈ V ( T ′ ).Let Q be the corridor of W corresponding to the edge vw . Since ˆ H is 3-connected, there is acycle O in W − E ( Q ) containing v and w . Let O and O denote the distinct cycles in O ∪ Q containing Q . Since γ ′ is shifting-equivalent to γ , from the assumption on W we deduce that γ ′ ( O ) , γ ′ ( O ) , γ ′ ( O ) ∈ Λ. Hence, 2 γ ′ ( Q ) = γ ′ ( O ) + γ ′ ( O ) − γ ′ ( O ) ∈ Λ and so σ (2 γ ′ ( Q )) is welldefined. Observe that 2 (cid:0) σ (2 γ ′ ( Q )) − γ ′ ( Q ) (cid:1) = 2 γ ′ ( Q ) − γ ′ ( Q ) = 0 . Let γ ′′ be the Γ-labelling of G obtained from γ ′ by shifting by σ (2 γ ′ ( Q )) − γ ′ ( Q ) at w . Then γ ′′ ( Q ) = γ ′ ( Q ) + σ (2 γ ′ ( Q )) − γ ′ ( Q ) = σ (2 γ ′ ( Q )) ∈ ˆΛ. Let T ′′ = T [ V ( T ′ ) ∪ { w } ]. Then γ ′′ ( P ) ∈ ˆΛfor all corridors P of W corresponding to edges of T ′′ , contradicting our choice of γ ′ and T ′ .Therefore T ′ = T . ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 15
Now, observe that γ ′ ( P ) ∈ ˆΛ for every corridor P of W , because Λ ⊆ ˆΛ and for every cycle O of W , we have γ ′ ( O ) = γ ( O ) ∈ Λ. Let γ ′′ be the ˆΛ-labelling of W which assigns an arbitraryedge e P of each corridor P the value γ ′ ( P ), and all other edges the value 0. By Lemma 5.3, thereis a ( w, w )-subwall W ′ of W which in particular is γ ′′ -bipartite, and hence γ ′ -bipartite. Since γ ′ and γ are shifting-equivalent, W ′ is γ -bipartite. (cid:3) Handling handles
This section is dedicated to proving the following key lemma, which allows us to iteratively findsets of vertex-disjoint handles.
Lemma 6.1.
There exist functions w . : N → N and f . : N → N satisfying the following. Let k, t and c be positive integers with c ≥ , let Γ be an abelian group, and let ( G, γ ) be a Γ -labelled graph.Let W be a wall in G of order at least w . ( k, c ) such that all corridors of W are γ -zero. Foreach i ∈ [ t − , let P i be a set of k W -handles in G such that the paths in S i ∈ [ t − P i are vertex-disjoint. If G contains at least f . ( k ) vertex-disjoint γ -non-zero V =2 ( W ) -paths, then there exist a c -column-slice W ′ of W and a set Q i of k vertex-disjoint W ′ -handles for each i ∈ [ t ] such that(i) for each i ∈ [ t − , the set Q i is a subset of the row-extension of P i to W ′ in W ,(ii) the paths in S i ∈ [ t ] Q i are vertex-disjoint,(iii) the paths in Q t are γ -non-zero. Before we can prove this lemma, we need to establish a variety of other lemmas. At the heart ofthe proof, we have the following natural result, which we will iteratively apply to decouple the setsof vertex-disjoint handles that we will construct. Huynh, Joos, and Wollan [10, Lemma 27] proveda somewhat similar result for oriented group-labelled graphs.
Lemma 6.2.
Let k, t be positive integers, let Γ be an abelian group, let ( G, γ ) be a Γ -labelled graph,and let T be a subset of V ( G ) . For each i ∈ [ t − , let P i be a set of T -paths of size k such thatthe paths in S i ∈ [ t − P i are vertex-disjoint.If G contains k vertex-disjoint γ -non-zero T -paths, then there exist a set Q t of k vertex-disjoint γ -non-zero T -paths and a subset Q i ⊆ P i of size k for each i ∈ [ t − so that the paths in S i ∈ [ t ] Q i are vertex-disjoint.Proof. Let Q t be a set of k vertex-disjoint γ -non-zero T -paths such that the number of edges ofpaths in Q t that are not contained in any path in P := S i ∈ [ t − P i is as small as possible. Foreach j ∈ [ t − P ∗ j be the set of paths in P j that do not contain an endvertex of a path in Q t .We have |P ∗ j | ≥ |P j | − |Q t | = 2 k . Let P ∗∗ j be the set of all paths in P ∗ j intersecting a path in Q t .Assume that for some j ∈ [ t −
1] we have |P ∗∗ j | ≥ k + 1. Then there are two paths P , P ∈ P ∗∗ j such that when traversing from an endvertex p i of P i for each i ∈ [2], P and P first meet thesame path Q ∈ Q t . For each i ∈ [2], let q i be the first intersection of P i and Q when traversing P i from p i .Let v and w be the endvertices of Q such that the distance between v and q in Q is smaller thanthe distance between v and q in Q . Let Q , Q , Q be the subpaths of Q from v to q , from q to q ,and from q to w , respectively. Also, for each i ∈ [2], let P ′ i be the subpath of P i from p i to q i , seeFigure 4. Since the paths of S i ∈ [ t − P i are vertex-disjoint and v, w / ∈ V ( P ∪ P ), both Q and Q contain edges not in a path of S i ∈ [ t − P i ; for instance, edges incident with q or q .By assumption, γ ( Q ) = γ ( Q ) + γ ( Q ) + γ ( Q ) is non-zero. If there is a { v, w, p , p } -path R in Q ∪ P ′ ∪ P ′ such that R = Q and γ ( R ) = 0, then by replacing Q with R , the number of edgesof paths in Q t that are not contained in any path in S i ∈ [ t − P i decreases. Therefore, by theassumption on Q t , we have γ ( R ) = 0 for every such path R . It implies that(1) γ ( Q ) + γ ( Q ) + γ ( P ′ ) = 0, p p v wP ′ P Q Q Q Q P ′ P q q Figure 4.
Segments of the paths P , P and Q mentioned in Lemma 6.2.(2) γ ( P ′ ) + γ ( Q ) + γ ( P ′ ) = 0,(3) γ ( P ′ ) + γ ( Q ) + γ ( Q ) = 0.The equations (1) and (2) imply that γ ( Q ) = γ ( P ′ ) and, similarly, the equations (2) and (3) implythat γ ( Q ) = γ ( P ′ ). But these imply that0 = γ ( P ′ ) + γ ( Q ) + γ ( P ′ ) = γ ( Q ) + γ ( Q ) + γ ( Q ) = 0 , which is a contradiction. We conclude for all j ∈ [ t −
1] that |P ∗∗ j | ≤ k , and thus |P ∗ j \ P ∗∗ j | ≥ k .For each j ∈ [ t − Q j be a set of k paths in P ∗ j \ P ∗∗ j . Then for each i ∈ [ t − Q i ⊆ P i , and the paths in S i ∈ [ t ] Q i are vertex-disjoint, as required. (cid:3) Lemma 6.1 mentions a set of vertex-disjoint V =2 ( W )-paths in G , but note that these may arbi-trarily intersect the internal vertices of corridors of W . The following technical lemma allows us totake subpaths of these paths which intersect the corridors of W in a more controlled manner. Lemma 6.3.
Let Γ be an abelian group, let ( G, γ ) be a Γ -labelled graph and let H ⊆ G be asubdivision of a -connected graph such that every corridor of H is γ -zero. If G contains a γ -non-zero V =2 ( H ) -path P , then there exist a subpath U of P and a set X of at most corridors of H satisfying the following properties:(i) H ∩ U is a subgraph of S X .(ii) For any subgraph H ′ ⊆ H which is a subdivision of a -connected graph with S X ⊆ H ′ ,and any subset T ⊆ V =2 ( H ′ ) with | T | ≥ , there is a γ -non-zero T -path in H ′ ∪ U .Proof. For each vertex z of H , we define x z, , x z, ∈ Γ and a path X z as follows. • If z has degree 2 in H , then let X z be the corridor of H containing z , let x z, := γ ( X z, ),and x z, := γ ( X z, ) where X z, and X z, are the two distinct subpaths of X z from z to theendvertices of X z . • Otherwise let X z be a path of length 0 containing z , let x z, := 0, and x z, := 0.A path Q from a ∈ V ( H ) to b ∈ V ( H ) in G with x a, = x a, and x b, = x b, is γ -preserving if x a, + γ ( Q ) + x b, = 0, and is γ -breaking otherwise. We first prove the following claim. Claim. P ∪ H contains a γ -breaking path U such that(a) both endvertices of U are in H ,(b) at most two corridors of H intersect the set of internal vertices of U , and(c) for each endvertex z of U , either z ∈ V =2 ( H ) or X z contains no internal vertex of U .Proof. Suppose that this claim does not hold. We first show that( ∗ ) if P contains a V ( H )-path Q from a to b where X a = X b , then Q is a γ -preserving path.Because there are no two distinct corridors of H with the same set of endvertices, X a inter-sects at most one of X b, and X b, . If γ ( X a, ) = γ ( X a, ) and X b,i does not intersect X a for ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 17 some i ∈ { , } , then X a, ∪ Q ∪ X b,i or X a, ∪ Q ∪ X b,i is γ -breaking. It is not difficult to ver-ify that such a γ -breaking path satisfies the required properties, contradicting the assumption.Thus, γ ( X a, ) = γ ( X a, ), and by symmetry, γ ( X b, ) = γ ( X b, ). Now, by the assumption, Q is γ -preserving. This shows ( ∗ ).Let M be the set of V ( H )-paths in P whose endvertices are internal vertices of distinct corridorsof H . Let M be the set of maximal subpaths of P − S Q ∈ M E ( Q ) of length at least 1. Note thatfor each R ∈ M , at most one corridor of H intersects the set of internal vertices of R .Let v and w be the endvertices of P , and let t := | M | + | M | . Note that M ∪ M is a partitionof P into t edge-disjoint subpaths, each having length at least 1. Let P be the path in M ∪ M containing v , and for each i ∈ [ t −
1] let P i +1 be the unique path in ( M ∪ M ) \ { P j : j ∈ [ i ] } sharing an endvertex, say v i , with P i . By ( ∗ ), we have γ ( X v i , ) = γ ( X v i , ) for all i ∈ [ t − γ ( X v, ) = γ ( X w, ) = 0.We claim that there is a γ -breaking path in M . Suppose for a contradiction that all paths in M are γ -preserving. By ( ∗ ), all paths in M are γ -preserving and therefore t − X i =0 (cid:0) γ ( X v i , ) + γ ( P i +1 ) + γ ( X v i +1 , ) (cid:1) = 0 . As every corridor of H is γ -zero, we know that t − X i =0 (cid:0) γ ( X v i , ) + γ ( X v i +1 , ) (cid:1) = 2 t − X i =1 γ ( X v i ) = 0 . This implies that P t − i =0 γ ( P i +1 ) = γ ( P ) = 0, which contradicts the fact that P is γ -non-zero. So,we conclude that there exists j ∈ [ t ] such that P j ∈ M and P j is γ -breaking.We obtain that the path P ′ defined by P ′ := P ∪ P if j = 1 ,P j − ∪ P j ∪ P j +1 if j ∈ [ t − \ { } ,P t − ∪ P t if j = t, has the desired properties. (cid:3) Let U be a path obtained by the previous claim. Let X be the set of corridors of H intersectingthe set of internal vertices of U . By the previous claim, we have |X | ≤ a , b be the endvertices of U . For x ∈ { a, b } , let Y x be a corridor of H containing x . Thusif x has degree 2 in H , then Y x = X x and otherwise Y x is an arbitrary corridor of H ending at x .Let X be a minimal set of corridors of H such that Y a , Y b ∈ X and each endvertex of Y a and Y b is contained in at least three corridors in X . Then |X | ≤ X := X ∪ X . Then |X | ≤
12 and (i) holds. It remains to show (ii). Let H ′ be a subgraphof H which is a subdivision of a 3-connected graph ˆ H ′ such that S X is a subgraph of H ′ and let T be subset of V =2 ( H ′ ) of size at least 3. Note that V =2 ( H ′ ) ⊆ V =2 ( H ) and therefore every corridorof H ′ is γ -zero. By the construction of X , each endvertex of U is contained in some corridor of H which is also a corridor of H ′ and every corridor of H intersecting the set of internal vertices of U is also a corridor of H ′ . Hence from the claim, we deduce that(a ′ ) both endvertices of U are in H ′ ,(b ′ ) at most two corridors of H ′ intersect the set of internal vertices of U , and(c ′ ) for each endvertex z of U , either z ∈ V =2 ( H ′ ) or the corridor of H ′ containing z containsno internal vertex of U .Since ˆ H ′ is 3-connected, there are two disjoint paths Q , Q in H ′ between the endvertices of U and the set T . If Q ∪ Q does not contain an internal vertex of U , then Q ∪ U ∪ Q is as desired,since U is γ -breaking and all corridors of H are γ -zero. If Q ∪ Q contains an internal vertex of U , then Q ∪ Q contains a corridor R of H ′ intersectingthe set of internal vertices of U . Choose x among two endvertices of R that is closer to T in Q ∪ Q .Then x is not an endvertex of U . Since x is in at least 3 corridors of H ′ , by property (b ′ ), x / ∈ V ( U ).Since ˆ H ′ is 3-connected, by properties (b ′ ) and (c ′ ), H ′ − E ( S X ) is connected. Thus, H ′ has a path Q connecting the endvertices of U which contains no internal vertex of U . The cy-cle O := Q ∪ U is γ -non-zero since U is γ -breaking. Since ˆ H ′ is 3-connected, there are threevertex-disjoint paths from T to { x } ∪ V =2 ( U ) in H ′ . By extending one of the paths ending at x to an internal vertex of U through R if x / ∈ V ( O ), we obtain three vertex-disjoint ( V ( O ) , T )-pathsin H ′ ∪ U . Hence Lemma 4.2 yields the desired result. (cid:3) In the next lemma, we extend subpaths from the previous lemma to handles of some suitablecolumn-slice.
Lemma 6.4.
There exist functions w . : N → N and f . : N → N satisfying the following. Let k and c be positive integers with c ≥ , let Γ be an abelian group, and let ( G, γ ) be a Γ -labelledgraph. Let W be a wall in G of order at least w . ( k, c ) such that all corridors of W are γ -zero.If G contains f . ( k ) vertex-disjoint γ -non-zero V =2 ( W ) -paths, then there exist a c -column-slice W ′ of W and k vertex-disjoint γ -non-zero W ′ -handles in G .Proof. Let h ( k ) := 3 f . ( k ) + 1 and f . ( k ) := 2 · h ( k ) . Let P be a set of f . ( k ) vertex-disjoint γ -non-zero V =2 ( W )-paths. Let w . ( k, c ) := (48 h ( k ) + 1)( c −
1) + 144 h ( k ) + 1. Claim.
For all i ∈ [ h ( k )] there exist a set C i of -column-slices of W , a set R i of -row-slices of W and a subpath U i of a path in P such that, with H i := S ( C i ∪ R i ) , we have(a) ≤ |C i | ≤ and ≤ |R i | ≤ ,(b) every C ∈ C i is vertex-disjoint from every C ′ ∈ C j for all j ∈ [ i − ,(c) every R ∈ R i is vertex-disjoint from every R ′ ∈ R j for all j ∈ [ i − ,(d) U i and U j are vertex-disjoint for all j ∈ [ i − ,(e) every -column-slice of W that intersects U i also intersects some -column-slice in S j ∈ [ i ] C j ,(f ) for any column-slice W ′ of W which is disjoint from U i , there is a γ -non-zero W ′ -handlein H i ∪ U i .Proof of Claim. We proceed by induction on i . For i ∈ [ h ( k )], assume that the claim holds forall j ∈ [ i − • e C i to be the set of all 3-column-slices of W which intersect no 3-column-slices in S j ∈ [ i − C j , • e R i to be the set of all 3-row-slices of W which intersect no 3-row-slices in S j ∈ [ i − R j , and • e H i := S (cid:16) e C i ∪ e R i (cid:17) .We will first show that the number of vertices in V =2 ( W ) \ V ( e H i ) is small. Let I be the set of allcolumn indices a such that the a -th column C Wa intersects no 3-column-slice in S j ∈ [ i − C j . Then I admits a partition into intervals consisting of consecutive integers such that the number of intervalsis bounded by P j ∈ [ i − |C j | + 1 ≤ i −
1) + 1. Observe that at least one interval of I has sizeat least 3 because w . ( k, c ) > · i −
1) + 2 · (48( i −
1) + 1). This implies that e C i is nonempty.Suppose that a vertex v in V =2 ( W ) is not in e H i . Let us say that v ∈ V ( C Wx ) ∩ V ( R Wy ) for some x and y . Since v is not in e H i , either C Wx intersects some 3-column-slice in C j for some j < i or x belongs to an interval of I of size at most 2. Since at least one interval of I has size more than 2,the number of possible values of x is at most3 · X j ∈ [ i − |C j | + 2 · X j ∈ [ i − |C j | ≤ i − . ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 19
By the same argument, we deduce that e R i is nonempty and the number of possible values of y is at most 240( i − V =2 ( W ) not in e H i is at most 2(240( i − ,because there are at most two vertices of V =2 ( W ) in V ( C Wx ) ∩ V ( R Wy ) for each x and y .Since |P| ≥ i ) > i − + ( i − P i in P both of whose endverticesare in e H i such that U j is not a subpath of P i for all j < i . Let X i be the set of at most 12 corridorsof e H i and let U i be a subpath of P i guaranteed by Lemma 6.3.Let C i be a minimal non-empty subset of e C i containing all 3-column-slices in e C i which intersectsome corridor in X i . Since each corridor of e H i intersects at most four 3-column-slices, |C i | ≤ · R i be a minimal non-empty subset of e R i containing all 3-row-slices in e R i whichintersect some corridor in X i . Then |R i | ≤
48. Now (a)–(e) are true by construction.To see (f), let W ′ be a column-slice disjoint from U i . Note that H i ∪ W ′ is a subdivision of a3-connected graph. Applying Lemma 6.3(ii) with H ′ := H i ∪ W ′ and T := V =2 ( H ′ ) ∩ V ( W ′ ), thereis a γ -non-zero T -path P in H ′ ∪ U i . Then P must use at least one edge of U i because γ ( P ′ ) = 0for every T -path P ′ in H ′ . This implies that P ⊆ H i ∪ U i , because T separates U i from W ′ in H ′ .It follows that P is a W ′ -handle. (cid:3) Let I be the set of all column indices a such that C Wa intersects no 3-column-slice in S j ∈ [ h ( k )] C j .By (a), I admits a partition into at most 48 h ( k ) + 1 disjoint intervals, each consisting of consecutiveintegers. Since w . ( k, c ) − · h ( k ) > (48 h ( k ) + 1)( c − c consecutive columns of W that do not intersect any 3-column-slice in S i ∈ [ h ( k )] C i . Thus they form a c -column-slice W ′ of W which is disjoint from S i ∈ [ h ( k )] U i by (e). By (f), for each i ∈ [ h ( k )], there is a γ -non-zero W ′ -handle X i in H i ∪ U i .By construction, each vertex of W − V ( W ′ ) is contained in at most two graphs in { H i : i ∈ [ h ( k )] } and in at most one path in { U i : i ∈ [ h ( k )] } . Hence, every vertex of W is contained in at most threepaths in { X i : i ∈ [ h ( k )] } . Since h ( k ) = 3 f . ( k ) + 1, any vertex set of size at most f . ( k ) cannot hitall these paths. By Theorem 2.9, there exist k vertex-disjoint γ -non-zero W ′ -handles, as desired. (cid:3) Now we obtain Lemma 6.1 as a corollary of Lemmas 6.2 and 6.4.
Proof of Lemma 6.1.
Let w . and f . be the functions w . and f . respectively, as given byLemma 6.4. By Lemma 6.4, there exists a c -column-slice W ′ of W and a set P ′ of k vertex-disjoint γ -non-zero W ′ -handles. For each i ∈ [ t −
1] and each P ∈ P i , let R P, and R P, be the rows of W containing the endvertices of P , and let Q P be the row-extension of P to W ′ , that is the unique V ( W ′ )-path such that P ⊆ Q P ⊆ P ∪ R P, ∪ R P, . We remark that it is possible that R P, = R P, .Now applying Lemma 6.2 to the sets { Q P : P ∈ P i } for i ∈ [ t −
1] together with P ′ yields thedesired result. (cid:3) Basic lemmas for products of abelian groups
In this section, we prove some basic lemmas on products of abelian groups that will be usefulthroughout Sections 8 and 9.An arithmetic progression is a set of integers A such that there are integers a and b = 0 forwhich A = { a + bn : n ∈ Z } . For a set A = { A i : i ∈ [ k ] } of arithmetic progressions, we say A covers a set S if S ⊆ S i ∈ [ k ] A i . We will use the following fact about arithmetic progressions, conjecturedby Erd˝os in 1962 and proven by Crittenden and Vanden Eynden [3] in 1969. We cite an equivalentversion of Balister et al. [1], who presented a simple proof. Theorem 7.1 ([3, 1]) . Let A = { A i : i ∈ [ k ] } be a set of k arithmetic progressions. If A covers aset of k consecutive integers, then A covers Z . Corollary 7.2.
Let m , t , and ω be positive integers, let Γ = Q j ∈ [ m ] Γ j be a product of m abeliangroups, and for all j ∈ [ m ] let Ω j be a subset of Γ j of size at most ω . For all i ∈ [ t ] and j ∈ [ m ] , let g i,j be an element of Γ j . If for each i ∈ [ t ] there exists an integer c i such that P ti =1 c i g i,j / ∈ Ω j for all j ∈ [ m ] , then for each i ∈ [ t ] there exists an integer d i ∈ [2 mω ] such that P ti =1 d i g i,j / ∈ Ω j forall j ∈ [ m ] .Proof. Pick an integer d i for each i ∈ [ t ] such that P ti =1 d i g i,j / ∈ Ω j for all j ∈ [ m ], and subject tothis |{ i ∈ [ t ] : d i ∈ [2 mω ] }| is maximised. Suppose for a contradiction that for some x ∈ [ t ], we havethat d x / ∈ [2 mω ]. Without loss of generality, we may assume x = t .For all j ∈ [ m ] and g ∈ Ω j , let A j,g be the set of integers d such that dg t,j + P t − i =1 d i g i,j = g . Notethat A j,g is an arithmetic progression or contains at most one integer. Let A ′ j,g be an arithmeticprogression such that A j,g ⊆ A ′ j,g and d t / ∈ A ′ j,g . Such an A ′ j,g exists; if A j,g is an arithmetic progres-sion then let A ′ j,g := A j,g , and if A j,g contains a unique integer a j , then let A ′ j,g be the arithmeticprogression { a j + 2( d t − a j ) k : k ∈ Z } .Now A := { A ′ j,g : j ∈ [ m ] , g ∈ Ω j } is a set of mω arithmetic progressions not covering d t . ByTheorem 7.1, there exists d ′ t ∈ [2 mω ] such that d ′ t g t,j + P t − i =1 d i g i,j / ∈ Ω j for all j ∈ [ m ], contradictingour choice of d t . (cid:3) For a sequence a = ( a i : i ∈ [ t ]) over an abelian group Γ, we let Σ( a ) denote the set of all sums ofsubsequences of a . We write | a | := t , the length of a . We say a ∈ Γ is repeated in a if a = a i = a j for some 1 ≤ i < j ≤ t . We say that a is good if | Σ( a ) | ≥ | a | . Obviously, a sequence of pairwisedistinct elements of Γ is good. Also, observe that for an element g of order at least t , if a i := g forall i ∈ [ t ], then the sequence a = ( a i : i ∈ [ t ]) is good, because | Σ( a ) | ≥ |{ kg : k ∈ [ t ] }| ≥ t . Lemma 7.3.
Let Γ be an abelian group and let a = ( a i : i ∈ [ t ]) be a sequence of length t over Γ .If all repeated elements in a have order at least t , then a is good.Proof. We proceed by induction on t . We may assume that t ≥
2. If a has no repeated ele-ments, then { a i : i ∈ [ t ] } ⊆ Σ( a ) and therefore a is good. Thus, without loss of generality, wemay assume that a t is a repeated element. Let a ′ := ( a i : i ∈ [ t − S := Σ( a ′ ). By induc-tion | S | ≥ t −
1. We may assume that | S | = t −
1. Let T := { x + a t : x ∈ S } ⊆ Σ( a ). If S = T ,then P x ∈ S x = P x ∈ S ( x + a t ) and therefore | S | a t = 0, contradicting the assumption on the orderof a t . Thus S = T and therefore | Σ( a ) | ≥ | S ∪ T | ≥ t . (cid:3) The following lemma and its corollary are useful to find a cycle whose γ i -value is not in Ω i forall i ∈ [ m ]. Lemma 7.4.
Let m , t , and ω be positive integers, let Γ = Q j ∈ [ m ] Γ j be a product of m abeliangroups and for all j ∈ [ m ] let Ω j be a subset of Γ j of size at most ω . For all i ∈ [ t ] let S i be a subsetof Γ such that for each j ∈ [ m ] there exists some i ∈ [ t ] such that π j ( g ) = π j ( g ′ ) for all distinct g, g ′ in S i . If | S i | > mω for all i ∈ [ t ] , then for every h ∈ Γ there is a sequence ( g i : i ∈ [ t ]) of elementsof Γ such that(i) g i ∈ S i for each i ∈ [ t ] , and(ii) π j (cid:16) h + P i ∈ [ t ] g i (cid:17) / ∈ Ω j for all j ∈ [ m ] .Proof. Uniformly at random, select g i ∈ S i independently for each i ∈ [ t ], and consider the sum g := h + P i ∈ [ t ] g i . For each j ∈ [ m ], there exists i ∈ [ t ] such that g and every group element obtainedby replacing g i in the sum with a different element of S i have distinct j -th coordinates. Hence, theprobability that π j ( g ) ∈ Ω j is at most ω/ ( mω + 1). It follows that there is a positive probabilitythat π j ( g ) / ∈ Ω j for all j ∈ [ m ]. (cid:3) For two sequences a = ( a i : i ∈ [ t ]), b = ( b i : i ∈ [ t ]) of length t over a product Γ = Q j ∈ [ m ] Γ j of m abelian groups, we write a − b to denote the sequence ( a i − b i : i ∈ [ t ]) and for j ∈ [ m ], wewrite π j ( a ) to denote the sequence ( π j ( a i ) : i ∈ [ t ]) over Γ j . ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 21
Corollary 7.5.
Let m and ω be positive integers, let Γ = Q i ∈ [ m ] Γ i be a product of m abelian groups,and for each i ∈ [ m ] let Ω i be a subset of Γ i of size at most ω and let a i := ( a i,j : j ∈ [ mω + 1]) and b i := ( b i,j : j ∈ [ mω + 1]) be two sequences over Γ such that π i ( a i − b i ) is good. Then forall h ∈ Γ , i ∈ [ m ] , and j ∈ [ mω + 1] , there exists c i,j ∈ { a i,j , b i,j } such that for all x ∈ [ m ] , wehave π x (cid:16) h + P i ∈ [ m ] P j ∈ [ mω +1] c i,j (cid:17) / ∈ Ω x .Proof. By definition of a good sequence, for each i ∈ [ m ], we have | Σ( π i ( a i − b i )) | ≥ mω + 1. Let S i be a subset of Σ( a i − b i ) such that π i restricted to S i is a bijection from S i to Σ( π i ( a i − b i )).Let h ′ := h + P i ∈ [ m ] P j ∈ [ mω +1] b i,j . We apply Lemma 7.4 with h ′ to find a sequence ( g i : i ∈ [ m ])such that g i ∈ S i for each i ∈ [ m ] and π j (cid:16) h ′ + P i ∈ [ m ] g i (cid:17) / ∈ Ω j for all j ∈ [ m ]. Now, for each i ∈ [ m ],we have that g i ∈ Σ( a i − b i ), and hence for each j ∈ [ mω + 1] there exists c i,j ∈ { a i,j , b i,j } suchthat g i + P j ∈ [ mω +1] b i,j = P j ∈ [ mω +1] c i,j . This completes the proof. (cid:3) Given positive integers n and k , we write R ( n ; k ) for the minimum integer N such that in every k -colouring of the edges of K N there is a monochromatic copy of K n . A classical result of Ramsey [15]shows that R ( n ; k ) exists. Lemma 7.6.
There exists a function f . : N → N satisfying the following. Let m , t and N bepositive integers with N ≥ f . ( t, m ) and let Γ = Q i ∈ [ m ] Γ i be a product of m abelian groups. Thenfor every sequence ( g i : i ∈ [ N ]) over Γ , there exists a subset I of [ N ] with | I | = t such that foreach i ∈ [ m ] , either • π i ( g j ) = π i ( g k ) for all j, k ∈ I , or • π i ( g j ) = π i ( g k ) for all distinct j, k ∈ I .Furthermore, if Z is a subset of [ m ] such that for all distinct i and j in [ N ] there exists x ∈ Z suchthat π x ( g i ) = π x ( g j ) , then the second condition holds for some i ∈ Z .Proof. Let f . ( t, m ) := R ( t ; 2 m ). We define a 2 m -colouring of the edges of K N by colouringeach edge xy of K N by the set { i ∈ [ m ] : π i ( g x ) = π i ( g y ) } . The result follows from the defini-tion of R ( t ; 2 m ). Note that if Z is subset of [ m ] such that for all distinct i and j in [ N ] there existsan x ∈ Z such that π x ( g i ) = π x ( g j ), then every set used in the colouring intersects Z . (cid:3) From handles to cycles
The focus of this section is proving the following key lemma, which will be the final ingredientneeded in the proof of Theorem 1.1 for constructing the cycles from the clean subwall from Section 5and the sets of handles from Section 6.
Lemma 8.1.
There exist functions c . , r . : N → N satisfying the following. Let t , ℓ , m and ω be positive integers with ℓ ≥ , let Γ = Q i ∈ [ m ] Γ i be a product of m abelian groups, for each i ∈ [ m ] let Ω i be a subset of Γ i of size at most ω , and let ( G, γ ) be a Γ -labelled graph. Let Z be a subsetof [ m ] , let W be a ( γ, Z, ℓ ) -clean ( c, r ) -wall with c ≥ c . ( t, ℓ, m, ω ) and r ≥ r . ( t, ℓ, m, ω ) . Forevery set P of at most t vertex-disjoint W -handles such that γ i ( S P ) / ∈ Ω i for all i ∈ Z , there is acycle O in W ∪ S P such that γ i ( O ) / ∈ Ω i for all i ∈ [ m ] . We begin by linking up a set of handles of a wall into a cycle.
Lemma 8.2.
Let t be a positive integer and let W be a ( c, r ) -wall in a graph G with r ≥ and c ≥ max { , t + 1 } . For every set P of at most t vertex-disjoint W -handles in G , there is acycle O in W ∪ S P that contains S P as a subgraph.Proof. Let T be the set of endvertices of all paths in P . We proceed by induction on t . If |P| ≤ W ∪ S P is 2-connected, W ∪ S P has a cycle O containing at least one edge from everypath in P . Therefore, we may assume that |P| = t > By symmetry, we may assume that the first column of W meets at least two paths in P . In thefirst column of W , choose two degree-2 nails v , v that are endvertices of distinct paths P , P of P respectively such that the distance between v and v in the first column of W is minimised.Let Q be the path from v to v in the first column of W . Let P ∗ be the path P ∪ Q ∪ P .Let W ′ be the ( c − W obtained by removing the first column. Let P ′ be therow-extension of ( P \ { P , P } ) ∪ { P ∗ } to W ′ . Since P ′ is a set of t − W ′ -handles,by the induction hypothesis, there is a cycle O in W ′ ∪ S P ′ such that S P ′ ⊆ O . This completesthe proof because S P ⊆ S P ′ and W ′ ∪ S P ′ ⊆ W ∪ S P . (cid:3) In order to obtain a cycle whose γ i -value is not in Ω i for every i ∈ [ m ], it will be useful to haveaccess to a sequence of subwalls to reroute segments of the cycle constructed by the previous lemma.The following straightforward corollary provides this. Corollary 8.3.
There exist functions c . , r . : N → N satisfying the following. Let t , k and w be positive integers with w ≥ . Let G be a graph containing a ( c, r ) -wall W with c ≥ c . ( t, k, w ) and r ≥ r . ( t, k, w ) . For every set P of at most t vertex-disjoint W -handles in G , there exist acycle O in W ∪ S P and a set { W i : i ∈ [ k ] } of k vertex-disjoint N W -anchored ( w, w ) -subwalls of W such that S P ⊆ O and W i ∩ O = R W i for all i ∈ [ k ] .Proof. Define c . ( t, k, w ) := kw + t + 1 , and r . ( t, k, w ) := (2 t + 1)( w −
2) + 1 . The case P = ∅ is easy to verify, so we may assume |P| = t >
0. Without loss of generality, we mayassume that the last column of W intersects S P . Let W ′′ be a kw -column-slice of W containingthe last column of W and let W ′ be a ( c − kw )-column-slice of W disjoint with W ′′ . Let P ′ denotethe row-extension of P to W ′ in W .By the pigeonhole principle, there is a ( w − W ′′ which is disjoint from S P ′ .Hence there is a w -row-slice S of W ′′ such that S ∩ S P ′ = R S . We can pack k vertex-disjoint N W -anchored ( w, w )-subwalls { W i : i ∈ [ k ] } in S so that W i ∩ S P ′ = R W i . Applying Lemma 8.2to W ′ and P ′ yields the desired cycle. (cid:3) Moreover, we need the following variation of Lemma 4.2.
Lemma 8.4.
Let Γ be an abelian group, and let ( G, γ ) be a Γ -labelled graph, let O be a γ -non-zerocycle in G , and let P be a path disjoint from O . If G contains three vertex-disjoint ( V ( P ) , V ( O )) -paths P , P , P , then there is a path P ′ in P ∪ O ∪ P ∪ P ∪ P with the same endvertices as P such that γ ( P ′ ) = γ ( P ) .Proof. Let T := V ( O ) ∩ ( V ( P ) ∪ V ( P ) ∪ V ( P )). Since | T | = 3, the cycle O contains three dis-tinct T -paths Q , Q , Q such that V ( Q i ) ∩ V ( P i ) = ∅ for each i ∈ [3]. Since γ ( O ) = 0, we havethat γ ( O ) = 2 γ ( O ), which implies γ ( Q ) + γ ( Q ) + γ ( Q ) = ( γ ( Q ) + γ ( Q )) + ( γ ( Q ) + γ ( Q )) + ( γ ( Q ) + γ ( Q )) . Without loss of generality, we may assume γ ( Q ) = γ ( Q ) + γ ( Q ). Observe that there are paths P ′ and P ′′ in P ∪ O ∪ P ∪ P with the same endvertices as P such that E ( P ′ ) \ E ( P ′′ ) = E ( Q )and E ( P ′′ ) \ E ( P ′ ) = E ( Q ) ∪ E ( Q ). Hence, P ′ or P ′′ is the desired path. (cid:3) Finally, we can prove Lemma 8.1.
Proof of Lemma 8.1.
For convenience, let w := w . ( mω, mω, ℓ ). We define c . ( t, ℓ, m, ω ) := max { , t + 1 , c . ( t, m ( mω + 1) , w + 1) } ,r . ( t, ℓ, m, ω ) := max { , r . ( t, m ( mω + 1) , w + 1) } . If Z = [ m ], then the result follows from Lemma 8.2. Hence, without loss of generality, we mayassume that Z = [ m ] \ [ y ] for some y ∈ [ m ]. By Corollary 8.3, there exist a cycle O in W ∪ S P and ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 23 a set { W i,j : i ∈ [ y ] , j ∈ [ yω + 1] } of y ( yω +1) vertex-disjoint N W -anchored ( w +1 , w +1)-subwallsof W such that S P ⊆ O and W i,j ∩ O = R W i,j =: P i,j for all i ∈ [ y ] and j ∈ [ yω + 1]. For i ∈ [ y ]and j ∈ [ yω + 1], let W ′ i,j be a w -row-slice of W i,j disjoint from O , and let H be the graph obtainedfrom O by deleting the internal vertices of the paths P i,j for all i ∈ [ y ] and j ∈ [ yω + 1].For each i ∈ [ y ], we now recursively define a family (cid:0) Q i,j : j ∈ [ yω + 1] (cid:1) of paths and a family( S i,j : j ∈ [ yω + 1]) of subsets of Γ i , such that for all j ∈ [ yω + 1] and g ∈ S i,j , • | S i,j | ≤ j −
1, and • the order of g is at most mω .We first set S i, := ∅ . Now, for j ∈ [ yω + 1], let λ j be the induced Γ i / h S i,j i -labelling of G . Notethat since i / ∈ Z and W is ( γ, Z, ℓ )-clean, W has no γ i -bipartite ( ℓ, ℓ )-subwall, and in particular,each W ′ i,j does not have such a subwall. As | S i,j | ≤ mω and each element of S i,j has order atmost mω , Corollary 5.4 implies that there is a λ j -non-zero cycle O i,j in W ′ i,j . By Lemma 8.4, thereis a path Q i,j in W i,j with the same endvertices as P i,j such that λ j ( P i,j ) = λ j ( Q i,j ), since thereare three vertex-disjoint ( V ( P i,j ) , V ( O i,j ))-paths in W i,j . We set S i,j +1 := ( S i,j if γ i ( Q i,j ) − γ i ( P i,j ) has order at least mω + 1 , and S i,j ∪ (cid:8) γ i ( Q i,j ) − γ i ( P i,j ) (cid:9) otherwise.For i ∈ [ y ], let D i := (cid:0) γ ( Q i,j ) − γ ( P i,j ) : j ∈ [ yω + 1] (cid:1) . By construction of the sets S i,j , the pro-jection of D i to Γ i contains no repeated elements of order at most mω in Γ i , and thus this sequenceis good by Lemma 7.3. Hence, by Corollary 7.5 with h := γ ( H ), there exists X i,j ∈ { P i,j , Q i,j } for all i ∈ [ y ] and j ∈ [ yω + 1] such that for the cycle O ′ := H ∪ S (cid:8) X i,j : i ∈ [ y ] , j ∈ [ yω + 1] (cid:9) , wehave that γ i ( O ′ ) / ∈ Ω i for all i ∈ [ y ]. By construction, γ j ( O ′ ) = γ j ( O ) / ∈ Ω j for all j ∈ Z because W is ( γ, Z, ℓ )-clean. (cid:3) Proof of the main theorem
We now complete the proof of Theorem 1.1, which we are restating for the convenience of thereader.
Theorem 1.1.
For every pair of positive integers m and ω , there is a function f m,ω : N → N satisfying the following property. For each i ∈ [ m ] , let Γ i be an abelian group, and let Ω i be a subsetof Γ i . Let G be a graph and for each i ∈ [ m ] , let γ i be a Γ i -labelling of G , and let O be the setof all cycles of G whose γ i -value is in Γ i \ Ω i for all i ∈ [ m ] . If | Ω i | ≤ ω for all i ∈ [ m ] , then forall k ∈ N there exists either a half-integral packing of k cycles in O , or a hitting set for O of sizeat most f m,ω ( k ) .Proof. Let Γ := Q i ∈ [ m ] Γ i denote the product of the m given abelian groups, and let γ : E ( G ) → Γdenote the Γ-labelling for which γ i ( e ) = π i ( γ ( e )) for all e ∈ E ( G ). We proceed by induction on k .For k ≤
2, we may trivially set f m,ω ( k ) = 1. Now suppose that k > f m,ω ( k −
1) as per the theorem. For every subgraph H of G , let ν ( H ) denote the maximumsize of a set of cycles O in H with γ i ( O ) / ∈ Ω i for all i ∈ [ m ] such that no three cycles in the setshare a common vertex. Observe that ν is a packing function for G . We will show that if τ ν ( G ) issufficiently large relative to k , then ν ( G ) ≥ k . This will complete the proof of the theorem.For non-negative integers p and z with z ≤ m , let α ( p, z ) and ρ ( z ) be recursively defined asfollows. For every non-negative integer p , we define α ( p,
0) := k mω + mω + 1 , ρ (0) := m + f . ( α (1 , , m ) , and for z > α ( p, z ) := 4 max { ρ ( z − − p, } α (1 , z − , ρ ( z ) := m + f . ( α (1 , z ) , m ) . Let ˆ p := ρ ( m − f . is increasing in its first argument, andhence ρ ( z ) ≤ ˆ p for z ≤ m −
1. Let u := max {⌈ f m,ω ( k − / ⌉ , f . ( f . ( f . ( α (1 , m ) , m ))) + 3 } . We recursively define β ( p, z , z ) for non-negative integers p , z , and z with z ≤ z ≤ m and p ≤ ˆ p ,as well as ψ ( z ) for a non-negative integer z with z ≤ m + 1 as follows. We define ψ ( m + 1) := 3 , and for z ≤ m we define β ( p, z , z ) := max (cid:8) u, kc . (2 mω ˆ p, ψ ( z + 1) + 2 , m, ω ) (cid:9) if z = 0 ,β (1 , z − , z ) if z > p = ˆ p, max (cid:8) β ( p + 1 , z , z ) , w . ( α ( p + 1 , z ) , β ( p + 1 , z , z )) (cid:9) if z > p < ˆ p ; ψ ( z ) := max (cid:8) ψ ( z + 1) , β (0 , z, z ) , r . (2 mω ˆ p, ψ ( z + 1) + 2 , m, ω ) (cid:9) . Observe that β ( p, z , z ) ≥ u .Lastly, we define f m,ω ( k ) := max (cid:8) f . ( ψ (0) + 2) , u, f m,ω ( k − (cid:9) .Let T be a minimum ν -hitting set of size t := τ ν ( G ), and assume that t > f m,ω ( k ). By theinduction hypothesis, G has a half-integral packing of k − O and therefore we mayassume for a contradiction that ν ( G ) = k −
1. For each subgraph H of G , if ν ( H ) < ν ( G ), thenby the induction hypothesis, τ ν ( H ) ≤ f m,ω ( k − ≤ f m,ω ( k ) / < t/
12. Lemma 4.1 yields thatthe set T T of all separations ( A, B ) of G of order less than t/ | B ∩ T | > t/ ⌈ t/ ⌉ > f . ( ψ (0) + 2). By Theorem 2.6, there is a ( ψ (0) + 2 , ψ (0) + 2)-wall in G dominatedby T T . By Lemma 5.1, this wall has a ( ψ ( | Z | ) , ψ ( | Z | ))-subwall W which is ( γ ′ , Z, ψ ( | Z | + 1) + 2)-clean for some subset Z ⊆ [ m ] and some Γ-labelling γ ′ of G shifting-equivalent to γ . By Lemma 2.8,the wall W is dominated by T T . Since γ ( O ) = γ ′ ( O ) for every cycle O in G , we may assume withoutloss of generality that γ = γ ′ . Claim 1.
There exist integers c and p with c ≥ β (1 , , z ) and ≤ p ≤ ˆ p , a c -column-slice W ′ of W ,a family P = ( P i : i ∈ [ p ]) of non-empty sets of W ′ -handles, and a family ( Z i : i ∈ { } ∪ [ p ]) ofdisjoint subsets of Z such that (a) if P ∈ P i and Q ∈ P j are not vertex-disjoint for some i, j ∈ [ p ] , then i = j and P = Q , (b) S i ∈{ }∪ [ p ] Z i = Z , (c) |P i | ≥ α ( p, | Z | ) for all i ∈ [ p ] , (d) | γ j ( P i ) | = |P i | for all i ∈ [ p ] and j ∈ Z i , (e) | γ j ( P i ) | = 1 for all i ∈ [ p ] and j ∈ Z , (f) there is some g ∈ h S i ∈ [ p ] γ ( P i ) i such that π j ( g ) / ∈ Ω j for all j ∈ Z .Proof. For non-negative integers c , q , and p , we say that a triple ( W ′ , P , Z ) consisting of a wall W ′ ,a family P := ( P i : i ∈ [ p ]) of non-empty sets of W ′ -handles, and a family Z := ( Z i : i ∈ { } ∪ [ p ])of disjoint subsets of Z is a ( c, q, p ) -McGuffin if W ′ is a c -column-slice of W , and W ′ , P , and Z satisfy (a)–(e), as well as the following two conditions:(g) Z i = ∅ for i ∈ [ q ] and Z i = ∅ for i ∈ [ p ] \ [ q ],(h) for all distinct i, i ′ ∈ [ p ] \ [ q ] there is j ∈ Z such that γ j ( P i ) ∩ γ j ( P i ′ ) = ∅ .Note that ( W, ∅ , ( Z )) is a ( ψ ( | Z | ) , , q, p ) be a lexicographically maximal pair ofnon-negative integers with q ≤ | Z | and q ≤ p ≤ ˆ p for which there is a ( c, q, p )-McGuffin ( W ′ , P , Z )for some c ≥ β ( p, | Z | , | Z | ). Now suppose for a contradiction that ( W ′ , P , Z ) does not satisfy (f).Then Z is nonempty. We distinguish two cases. ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 25 If p = ˆ p , then p − q ≥ ˆ p − m ≥ ρ ( z ) − m ≥ f . ( α ( q + 1 , | Z | − , m ) since q ≤ m by (g) and α is decreasing in its first argument. Let P ′′ be a set of p − q vertex-disjoint W ′ -handles contain-ing exactly one element of P i for each i ∈ [ p ] \ [ q ]. Such a set P ′′ exists by (a). For i ∈ [ q ],let P ′ i := P i and Z ′ i := Z i . Observe that by (h), for all distinct paths P and Q in P ′′ , thereexists j ∈ Z such that γ j ( P ) = γ j ( Q ). Thus, by Lemma 7.6, there is a subset P ′ q +1 of P ′′ with |P ′ q +1 | = α ( q + 1 , | Z | −
1) such that for each i ∈ [ m ], either • γ i ( P ) = γ i ( Q ) for all P, Q ∈ P ′ q +1 , or • γ i ( P ) = γ i ( Q ) for all distinct P, Q ∈ P ′ q +1 ,and the second condition holds for some i ∈ Z . Let Z ′ q +1 := { i ∈ Z : γ i ( P ) = γ i ( Q ) for all distinct P, Q ∈ P ′ q +1 } and Z ′ := Z \ Z ′ q +1 . Let P ′ := ( P ′ i : i ∈ [ q + 1]) and Z ′ := ( Z ′ i : i ∈ { } ∪ [ q + 1]). Then ( W ′ , P ′ , Z ′ ) is a ( c, q + 1 , q + 1)-McGuffin, since (c) follows from the fact that |P ′ q +1 | ≥ α ( q + 1 , | Z | − ≥ α ( q + 1 , | Z ′ | ), and theremaining conditions ((a), (b), (d), (e), (g), and (h)) are easy to check. This contradicts themaximality of ( q, p ).So we may assume that p < ˆ p . Let Λ be the subgroup of Γ consisting of all g ∈ Γ for which thereis g ′ ∈ h S i ∈ [ p ] γ ( P i ) i such that for all j ∈ Z we have π j ( g ) = π j ( g ′ ). Let λ be the induced Γ / Λ-labelling of G . Note that by the negation of (f), neither h S i ∈ [ p ] γ ( P i ) i nor Λ contains an element g such that π j ( g ) / ∈ Ω j for all j ∈ Z . Therefore,(1) every cycle O of G for which γ i ( O ) / ∈ Ω i for all i ∈ [ m ] is λ -non-zero.Note that W ′ is a subwall of W of order c ≥ u . For any S ⊆ V ( G ) of size at most u − X of G − S containing a row of W ′ . By Lemma 2.8, T T dominates W ′ ,so the separation ( V ( G ) \ V ( X ) , S ∪ V ( X )) is in T T , so X contains a vertex of V =2 ( W ′ ) and atleast 5 t/ − ( u − > f m,ω ( k ) / − ( u − > u vertices of T . By (1), every minimal subgraph H with ν ( H ) ≥ λ -non-zero cycle. Moreover, if H is a subgraph of G with ν ( H ) < ν ( G ) = k − τ ν ( H ) ≤ f m,w ( k − ≤ u . Hence, by Lemma 4.3, we have that G contains a set of f . ( f . ( α (1 , m ) , m )) disjoint λ -non-zero V =2 ( W ′ )-paths. We may assume thatfunction w . is increasing in both of its arguments. As | Z | > p < ˆ p , we have c ≥ β ( p, | Z | , | Z | ) ≥ w . ( α ( p + 1 , | Z | ) , β ( p + 1 , | Z | , | Z | )) . Thus, by Lemma 6.1 applied to W ′ , there exist a c ′ -column-slice W ′′ of W ′ for some c ′ ≥ β ( p + 1 , | Z | , | Z | ) ≥ β ( q + 1 , | Z | − , | Z | )and a set P ′ i of f . ( α ( p + 1 , | Z | ) , m ) vertex-disjoint W ′′ -handles for each i ∈ [ p + 1] such that • for each i ∈ [ p ], the set P ′ i is a subset of the row-extension of P i to W ′′ in W ′ , • the paths in S i ∈ [ p +1] P ′ i are vertex-disjoint, • the paths in P ′ p +1 are λ -non-zero.By Lemma 7.6, there exist a subset R of P ′ p +1 and a subset Z ′ of Z such that • | γ j ( R ) | = |R| for all j ∈ Z ′ , • | γ j ( R ) | = 1 for all j ∈ Z \ Z ′ , and • |R| ≥ α ( p + 1 , | Z | ) ≥ α ( q + 1 , | Z | − p ′′ := p + 1 if Z ′ is empty and let p ′′ := q + 1 if Z ′ is non-empty, and For i ∈ { } ∪ [ p ′′ ], let Z ′′ i := Z \ Z ′ if i = 0 ,Z i if i ∈ [ p ′′ − ,Z ′ if i = p ′′ . For i ∈ [ p ′′ − P ′′ i := P ′ i and let P p ′′ := R . We now show that (cid:0) W ′′ , ( P ′′ i : i ∈ [ p ′′ ]) , ( Z ′′ i : i ∈ { } ∪ [ p ′′ ]) (cid:1) is either a ( c ′ , q, p + 1)-McGuffin ora ( c ′ , q + 1 , q + 1)-McGuffin, contradicting the maximality of ( q, p ). First, observe that since W is ( γ ′ , Z, ψ ( | Z | + 1) + 2)-clean, every N W -path is γ i -zero for all i ∈ Z and therefore if P ′ is therow-extension of P to W ′′ in W ′ , then γ i ( P ′ ) = γ i ( P ) for all i ∈ Z , implying (d) and (e) for i < p ′′ .By the definition of Z ′ , properties (d) and (e) hold for i = p ′′ . It remains to check (h) when Z ′ isempty, q < i ≤ p , and i ′ = p ′′ = p + 1. This is implied by the property that the paths in P ′ p +1 are λ -non-zero. (cid:3) Claim 2.
There is a family ( Q i : i ∈ [ k ]) of k disjoint subsets of S i ∈ [ p ] P i , each of size at most | Z | ω p , such that for each i ∈ [ k ] and each j ∈ Z , we have γ j (cid:0) S Q i (cid:1) / ∈ Ω j .Proof. Recursively, for each i ∈ [ k ] we define Q i containing at most 2 | Z | ω elements of P j forall j ∈ [ p ]. For each i ∈ [ k ] and j ∈ [ p ], let X i,j := P j ∩ S i ′ ∈ [ i − Q i ′ , and note that we have X ,j = ∅ and |X i,j | ≤ ( i − | Z | w ≤ ( k − mw .For each j ∈ [ p ], select g j ∈ γ ( P j ) arbitrarily. By Claim 1(e) and (f), for each j ∈ [ p ] there existsan integer c j such that π x (cid:0) P j ∈ [ p ] c j g j (cid:1) / ∈ Ω x for all x ∈ Z . Hence, by Corollary 7.2, for each j ∈ [ p ]there exists an integer d j ∈ (cid:2) | Z | ω (cid:3) such that for all x ∈ Z , we have π x (cid:0) P j ∈ [ p ] d j g j (cid:1) / ∈ Ω x . Let I be the set of indices j ∈ [ p ] such that Z j = ∅ . Now |P j | ≥ α ( p, | Z | ) ≥ α ( p, ≥ k mw ≥ |X i,j | + d j for all j ∈ [ p ] by Claim 1(c). Hence, for each j ∈ [ p ], we can select a set Y j of distinct W ′ -handles in P j \ X i,j of size d j − j ∈ I and of size d j otherwise. By the definition of X i,j andClaim 1(c), there are at least α ( p, | Z | ) − k | Z | ω ≥ mω + 1 distinct W ′ -handles in P j \ ( X i ∪ Y j )for every j ∈ [ p ]. Define h := X j ∈ [ p ] X P ∈Y j γ ( P ) , and for j ∈ I define S j := { γ ( P ) : P ∈ P j \ ( X i,j ∪ Y j ) } . By Claim 1(d), for all j ∈ I , for alldistinct g , g ′ in S j , and for all x ∈ Z j , we have π x ( g ) = π x ( g ′ ) and so | S j | > mw . Now byLemma 7.4, there is a family ( Q j : j ∈ I ) of paths such that Q j ∈ P j \ ( X i,j ∪ Y j ) for all j ∈ I ,and π x (cid:0) h + P j ∈ I γ ( Q j ) (cid:1) / ∈ Ω x for all x ∈ Z . Hence, let Q i := { Q j : j ∈ I } ∪ S j ∈ [ p ] Y j , and notethat |Q i | ≤ P j ∈ [ p ] d j ≤ | Z | w p . (cid:3) We now complete the proof of the theorem. Since W ′ has at least β (1 , , | Z | ) columns, there isa set { W i : i ∈ [ k ] } of k vertex-disjoint c . (2 mω ˆ p, ψ ( | Z | + 1) + 2 , m, ω )-column-slices of W ′ . Notethat the number of rows of W ′ is at least ψ ( | Z | ) ≥ r . (2 mω ˆ p, ψ ( | Z | + 1) + 2 , m, ω ). For each i ∈ [ k ],let Q ∗ i be the row-extension of Q i to W i . By Lemma 8.1, for each i ∈ [ k ] there is a cycle O i in W i ∪ S Q ∗ i with γ j ( O i ) / ∈ Ω j for all j ∈ [ m ]. Observe that for every vertex v ∈ V ( G ), there areat most two indices i ∈ [ k ] such that v ∈ V ( W i ∪ S Q ∗ i ). Hence, ν ( G ) ≥ k , a contradiction. (cid:3) Conclusion
In this work, we proved that a half-integral analogue of the Erd˝os-P´osa theorem holds for cyclesin graphs labelled with a bounded number of abelian groups, whose values avoid a bounded numberof elements of each group. We conclude with some open problems.In the proof of our theorem, the theorem of Wollan [28] about γ -non-zero A -paths was important.This theorem implies that an analogue of the Erd˝os-P´osa theorem holds for the odd A -paths, andfor A -paths intersecting a prescribed set of vertices. Bruhn, Heinlein, and Joos [2] further showedthat an analogue of the Erd˝os-P´osa theorem holds for A -paths of length at least ℓ , and for A -paths of even length, but interestingly, they also showed that for every composite integer m > d ∈ { } ∪ [ m − A -paths of length d modulo m . Later,Thomas and Yoo [24] characterised the abelian groups Γ and elements ℓ ∈ Γ where an analogue ofthe Erd˝os-P´osa theorem holds for A -paths of γ -value ℓ . We would like to ask whether a statementsimilar to Theorem 1.1 holds for A -paths. ALF-INTEGRAL ERD ˝OS-P ´OSA FOR CYCLES IN GRAPHS LABELLED BY MULTIPLE ABELIAN GROUPS 27
Question 1.
For every pair of positive integers m and ω , does there exist a function g m,ω : N → N satisfying the following property? • For each i ∈ [ m ] , let Γ i be an abelian group and let Ω i be a subset of Γ i . Let G be a graph,let A be a set of vertices in G , and for each i ∈ [ m ] , let γ i be a Γ i -labelling of G , and let P be the set of all A -paths of G whose γ i -value is in Γ i \ Ω i for all i ∈ [ m ] . If | Ω i | ≤ ω forall i ∈ [ m ] , then there exists either a half-integral packing of k paths in P , or a hitting setfor P of size at most g m,ω ( k ) . Our next question relates to directed labellings of graphs. Let Γ be a group (not necessar-ily abelian). A directed Γ -labelling of a graph G is a function γ from the set of oriented edges ~E ( G ) := { ( e, w ) : e = uv ∈ E ( G ) , w ∈ { u, v }} to Γ such that γ ( e, u ) = − γ ( e, v ) for each edge e = uv .Given a walk W := v t e v e · · · e t v t , we define γ ( W ) := P tj =1 γ ( e j , v j ), and say that W corresponds to a cycle O if E ( O ) = E ( W ) and v t is the only repeated vertex of W . It is straightforward to checkthat if any walk corresponding to a cycle O has value 0, then all walks corresponding to O do, soit makes sense to consider non-zero cycles with respect to a directed labelling. Note that if Γ isabelian and W and W are walks corresponding to the same cycle O , then γ ( O ) = ± γ ( O ). If Γ isnot abelian, then the choice of start vertex for the corresponding walk does matter as well. Hence,in this case, we are really considering cycles together with a specified start vertex and direction.Huynh, Joos, and Wollan [10] conjectured that a half-integral analogue of the Erd˝os-P´osa theoremholds for cycles which are non-zero with respect to a fixed number of directed labellings. We askwhether a statement similar to Theorem 1.1 holds for directed labellings. If it does, then it wouldimply the conjecture of Huynh, Joos, and Wollan. Question 2.
For every pair of positive integers m and ω , does there exist a function g m,ω : N → N satisfying the following property? • For each i ∈ [ m ] , let Γ i be a group and let Ω i be a subset of Γ i . Let G be a graph and foreach i ∈ [ m ] , let γ i be a directed Γ i -labelling of G , and let O be the set of all cycles of G which have a corresponding walk W such that γ i ( W ) is in Γ i \ Ω i for all i ∈ [ m ] . If | Ω i | ≤ ω for all i ∈ [ m ] , then there exists either a half-integral packing of k cycles in O , or a hittingset for O of size at most g m,ω ( k ) . As discussed in Section 3.2, an analogue of the Erd˝os-P´osa theorem does not hold for the cyclesdescribed in Theorem 1.1. It was shown that in graphs of sufficiently high connectivity [27, 16,13, 11], an analogue of the Erd˝os-P´osa theorem holds for odd cycles. We ask whether a similarphenomenon happens for the cycles described in Theorem 1.1.
Question 3.
For every pair of positive integers m and ω , do there exist functions g m,ω : N → N and g ′ m,ω : N → N satisfying the following property? • For each i ∈ [ m ] , let Γ i be an abelian group and let Ω i be a subset of Γ i . Let G be a graph,and for each i ∈ [ m ] , let γ i be a Γ i -labelling of G , and let O be the set of all cycles of G whose γ i -value is in Γ i \ Ω i for all i ∈ [ m ] . If G is g ′ m,ω ( k ) -connected and | Ω i | ≤ ω for all i ∈ [ m ] ,then there exists either a set of k vertex-disjoint cycles in O , or a hitting set for O of sizeat most g m,ω ( k ) . Similar to Question 3, we ask whether an analogue of the Erd˝os-P´osa theorem holds for the A -paths described in Question 1, and for the cycles described in Question 2, in the case of highlyconnected graphs. References [1] P. Balister, B. Bollob´as, R. Morris, J. Sahasrabudhe, and M. Tiba. Covering intervals with arithmetic progres-sions.
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Combinatorica , 31(1):95–126, 2011. 1, 1, 3.2(Gollin, Hendrey, Kwon, Oum)
Discrete Mathematics Group, Institute for Basic Science (IBS),Daejeon, South Korea. (Kawarabayashi)
National Institute of Informatics, 2-1-2, Hitotsubashi, Chiyoda-ku, Tokyo, Japan (Kwon)
Department of Mathematics, Incheon National University, Incheon, South Korea. (Oum)
Department of Mathematical Sciences, KAIST, Daejeon, South Korea.
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